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Q:

In a race, Jonathan placed \(10\) distance markers along the running route. The distance between the first and second distance marker is \(150 m\) . What is the distance between the first and the last distance marker? (Hint: Draw a diagram)

Q:

(a) Let \(P Q R S\) be the quadrilateral with vertices \(P ( 1,0 ) , Q ( 0,2 ) , R ( - 3,0 )\) and \(S ( 0 , - 4 )\) . (i) Find \(P Q ^ { 2 } , R S ^ { 2 } , P S ^ { 2 }\) and \(Q R ^ { 2 }\) . (ii) Show that \(P Q ^ { 2 } + R S ^ { 2 } = P S ^ { 2 } + Q R ^ { 2 }\) .

(b) THEOREM: If the diagonals of a quadrilateral are perpendicular, then the two sums of squares of opposite sides are equal. Prove this theorem for any quadrilateral by placing the vertices on the axes, giving them coordinates \(P ( p , 0 ) , Q ( 0 , q ) , R ( - r , 0 )\) and \(S ( 0 , - s )\) , and proceeding as in part (a).

Q:

(a) The points \(O ( 0,0 ) , P ( 8,0 )\) and \(Q ( 0,10 )\) form a right- angled triangle. Let \(M\) be the midpoint of \(P Q\) . (i) Find the coordinates of \(M\) . (ii) Find the distances \(O M , P M\) and \(Q M\) , and show that \(M\) is equidistant from each of the vertices. (iii) Explain why a circle with centre \(M\) can be drawn through the three vertices \(O , P\) and \(Q\)

(b) THEOREM: The midpoint of the hypotenuse of a right-angled triangle is the centre of a circle through all three vertices. Prove this theorem for any right-angled triangle by placing its vertices at \(O ( 0,0 )\) , \(P ( 2 p , 0 )\) and \(Q ( 0,2 q )\) and repeating the procedures of part (a).

Q:

A ship sailed \(140\) nautical miles from port \(P\) to port \(Q\) on a bearing of \(050 ^ { \circ } T\) . It then sailed \(260\) nautical miles from port \(Q\) to port \(R\) on a bearing of \(130 ^ { \circ } T\) .

(a) Explain why \(\angle P Q R = 100 ^ { \circ }\) .

(b) Find the distance between ports \(R\) and \(P\) , correct to the nearest nautical mile.

(c) Find the bearing of port \(R\) from port \(P\) , correct to the nearest degree.

Q:

A ladder of length \(3\) metres is leaning against a wall and is inclined at \(62 ^ { \circ }\) to the ground. How far does it reach up the wall? (Answer in metres correct to two decimal places.)

Q:

A ship sails \(53\) nautical miles from \(P\) to \(Q\) on a bearing of \(026 ^ { \circ } T\) . It then sails \(78\) nautical miles due east from \(Q\) to \(R\) .

(a) Explain why \(\angle P Q R = 116 ^ { \circ }\) .

(b) How far apart are \(P\) and \(R\) , correct to the nearest nau- tical mile?

Q:

A boat is \(200\) metres out to sea from a vertical cliff of height \(40\) metres. Find, correct to the nearest degree, the angle of depression of the boat from the top of the cliff.

Q:

(a) A triangle has vertices at \(A ( 1 , - 3 ) , B ( 3,3 )\) and \(C ( - 3,1 )\) . (i) Find the coordinates of the midpoint \(P\) of \(A B\) and the midpoint \(Q\) of \(B C\) . (ii) Show that \(P Q \| A C\) and that \(P Q = \frac { 1 } { 2 } A C\) .

(b) THEOREM: The interval joining the midpoints of two sides of a triangle is parallel to the base and half its length. Prove this theorem for any triangle by placing its vertices at \(A ( 2 a , 0 ) , B ( 2 b , 2 c )\) and \(C ( 0,0 )\) , where \(a > 0\) , and proceeding as in part (a).

Q:

Find, correct to the nearest tenth of a metre, the height of a tower, if the angle of elevation of the top of the tower is \(64 ^ { \circ } 48 ^ { \prime }\) from a point on horizontal ground \(10\) metres from the base of the tower.

Q:

A ship is sailing at \(15 km / h\) on a bearing of \(160 ^ { \circ } T\) . At 9:00 am it is at \(P\) , and lighthouse \(L\) is due south. At \(9 : 40 am\) it is at \(Q\) , and the lighthouse is on a bearing of \(230 ^ { \circ } T\) .

(a) Show that \(\angle P Q L = 110 ^ { \circ }\) .

(b) Find the distance \(P L\) , correct to the nearest kilometre.

(c) Find the time, correct to the nearest minute, at which the lighthouse will be due west of the ship.

Q:

Each set of three points given below forms a triangle of one of these types: A. isosceles, B. equilateral, C. right-angled, D. none of these.

Q:

(a) A circle with centre \(O ( 0,0 )\) passes through \(A ( 5,12 )\) . What is its radius?

(b) A circle with centre \(B ( 4,5 )\) passes through the origin. What is its radius?

(c) Find the centre of the circle with diameter \(C D\) , where \(C = ( 2,1 )\) and \(D = ( 8 , - 7 )\) .

(d) Find the radius of the circle with diameter \(C D\) in part (c) above. (e) Show that \(E ( - 12 , - 5 )\) lies on the circle with centre the origin and radius \(13\) .

Q:

Find, in general form, the equation of the line:

(a) with gradient \(- 2\) and \(y\) -intercept \(5\) ,

(b) with gradient \(\frac { 2 } { 3 }\) through the point \(A ( 3,5 )\)

(c) through the origin perpendicular to \(y = 7 x - 5\) ,

(d) through \(B ( - 5,7 )\) parallel to \(y = 4 - 3 x\) (e) with \(y\) -intercept \(- 2\) and angle of inclination \(60 ^ { \circ }\) .

Q:

Prove this theorem for any triangle by placing its vertices at \(A ( 2 a , 0 ) , B ( - 2 a , 0 )\) and \(C ( 2 b , 2 c )\) , and proceeding as follows.

(a) Find the gradients of \(A B , B C\) and \(C A\) .

(b) Hence find the equations of the three perpendicular bisectors.

(c) Find the intersection \(M\) of any two, and show that it lies on the third.

(d) Explain why the circumcentre must be equidistant from each vertex.

Q:

(a) Write down two points \(A\) and \(B\) for which the interval \(A B\) has gradient \(3\) .

(b) Write down two points \(A\) and \(B\) for which the interval \(A B\) is vertical.

(c) Write down two points \(A\) and \(B\) for which \(A B\) has gradient \(2\) and midpoint \(M ( 4,6 )\) .

Q:

(a) Find the point \(M\) of intersection of the lines \(\ell _ { 1 } : x + y = 2\) and \(\ell _ { 2 } : 4 x - y = 13\) .

(b) Show that \(M\) lies on \(\ell _ { 3 } : 2 x - 5 y = 11\) , and hence that \(\ell _ { 1 } , \ell _ { 2 }\) and \(\ell _ { 3 }\) are concurrent.

(c) Use the same method to test whether each set of lines is concurrent. (i) \(2 x + y = - 1 , x - 2 y = - 18\) and \(x + 3 y = 15\) (ii) \(6 x - y = 26,5 x - 4 y = 9\) and \(x + y = 9\)

Q:

In each part below, the angle of inclination \(\alpha\) and the \(y\) -intercept \(A\) of a line are given. Use the formula gradient \(= \tan \alpha\) to find the gradient of each line, then find its equation in general form.

(a) \(\alpha = 45 ^ { \circ } , A = ( 0,3 )\)

(c) \(\alpha = 30 ^ { \circ } , A = ( 0 , - 2 )\)

(b) \(\alpha = 60 ^ { \circ } , A = ( 0 , - 1 )\)

(d) \(\alpha = 135 ^ { \circ } , A = ( 0,1 )\)

Q:

A ship sails \(50 km\) from port \(A\) to port \(B\) on a bearing of \(063 ^ { \circ } T\) , then sails \(130 km\) from port \(B\) to port \(C\) on a bearing of \(296 ^ { \circ } T\) .

(a) Show that \(\angle A B C = 53 ^ { \circ }\) .

(b) Find, correct to the nearest \(km\) , the distance of port \(A\) from port \(C\) .

(c) Use the cosine rule to find \(\angle A C B\) , and hence find the bearing of port \(A\) from port \(C\) , correct to the nearest

Q:

A golfer at \(G\) played a shot that landed \(10\) metres from the hole \(H\) . The direction of the shot was \(7 ^ { \circ }\) away from the direct line between \(G\) and \(H\) .

(a) Find, correct to the nearest minute, the two possible sizes of \(\angle G B H\) .

(b) Hence find the two possible distances the ball has trav- elled. (Answer in metres correct to one decimal place.)

Q:

Answer correct to four significant figures, or correct to the nearest minute.

(a) A triangle has sides of \(7 cm , 7 cm\) and \(5 cm\) . What are the sizes of its angles?

(b) A rectangle has dimensions \(7 cm \times 12 cm\) . At what acute angle do the diagonals meet?

(c) The diagonals of a rhombus are \(16 cm\) and \(10 cm\) . Find the vertex angles.

Q:

In a triangle \(X Y Z , \angle Y = 72 ^ { \circ }\) and \(\angle Y X Z = 66 ^ { \circ } . X P \perp Y Z\) and \(X P = 25 cm\) .

(a) Use the sine ratio in \(\triangle P X Y\) to show that \(X Y \div 26.3 cm\) .

(b) Hence use the sine rule in \(\triangle X Y Z\) to find \(Y Z\) , correct to the nearest centimetre.

(c) Check your answer to part (b) by using the tangent ratio in triangles \(P X Y\) and \(P X Z\) to find \(P Y\) and \(P Z\) .

Q:

Consider the two lines \(\ell _ { 1 } : 3 x + 2 y + 4 = 0\) and \(\ell _ { 2 } : 6 x + \mu y + \lambda = 0\) .

(a) Write down the value of \(\mu\) if: (i) \(\ell _ { 1 }\) is parallel to \(\ell _ { 2 }\) , (ii) \(\ell _ { 1 }\) is perpendicular to \(\ell _ { 2 }\) .

(b) Given that \(\ell _ { 1 }\) and \(\ell _ { 2 }\) intersect at a point, what condition must be placed on \(\mu\) ?

(c) Given that \(\ell _ { 1 }\) is parallel to \(\ell _ { 2 }\) , write down the value of \(\lambda\) if: (i) \(\ell _ { 1 }\) is the same line as \(\ell _ { 2 }\) , (ii) the distance between the \(y\) -intercepts of the two lines is \(2\) .

Q:

The point \(M ( 3,7 )\) is the midpoint of the interval joining \(A ( 1,12 )\) and \(B ( x _ { 2 } , y _ { 2 } )\) . Find the coordinates \(x _ { 2 }\) and \(y _ { 2 }\) of \(B\) by substituting into the formulae \(x = \frac { x _ { 1 } + x _ { 2 } } { 2 }\) and \(y = \frac { y _ { 1 } + y _ { 2 } } { 2 }\)

Q:

Find the gradient of the line through each pair of given points. Then find its equation, using gradient-intercept form. Give your final answer in general form.

(a) \(( 0,4 ) , ( 2,8 )\)

(b) \(( 0,0 ) , ( 1 , - 1 )\)

(c) \(( - 9 , - 1 ) , ( 0 , - 4 )\)

(d) \(( 2,6 ) , ( 0,11 )\)

Q:

Let the vertices of the quadrilateral be \(A ( a _ { 1 } , a _ { 2 } ) , B ( b _ { 1 } , b _ { 2 } )\) , \(C ( c _ { 1 } , c _ { 2 } )\) and \(D ( d _ { 1 } , d _ { 2 } )\) , as in the diagram opposite.

(a) Find the midpoints \(P , Q , R\) and \(S\) of the sides \(A B , B C\) , \(C D\) and \(D A\) respectively. (The figure \(P Q R S\) is also a quadrilateral.)

(b) Find the midpoints of the diagonals \(P R\) and \(Q S\) .

(c) Explain why this proves that \(P Q R S\) is a parallelogram.

Q:

The interval joining \(A ( 2 , - 5 )\) and \(E ( - 6 , - 1 )\) is divided into four equal subintervals by the three points \(B , C\) and \(D\) .

(a) Find the coordinates of \(C\) by taking the midpoint of \(A E\) .

(b) Find the coordinates of \(B\) and \(D\) by taking the midpoints of \(A C\) and \(C E\) .

Q:

The diagram shows three straight roads, \(A B , B C\) and \(C A\) , where \(A B = 8.3 km , A C = 15 \cdot 2 km\) , and the roads \(A B\) and \(A C\) intersect at \(57 ^ { \circ }\) . Two cars, \(P _ { 1 }\) and \(P _ { 2 }\) , leave \(A\) at the same instant. Car \(P _ { 1 }\) travels along \(A B\) and then \(B C\) at \(80 km / h\) while \(P _ { 2 }\) travels along \(A C\) at \(50 km / h\) . Which car reaches \(C\) first, and by how many minutes? (Answer correct to one decimal place.)

Q:

It is given that \(\alpha\) is an acute angle and that \(\tan \alpha = \frac { \sqrt { 5 } } { 2 }\) .

(a) Draw a right-angled triangle showing this information.

(b) Use Pythagoras' theorem to find the length of the unknown side.

(c) Hence write down the exact values of \(\sin \alpha\) and \(\cos \alpha\) .

(d) Show that \(\sin ^ { 2 } \alpha + \cos ^ { 2 } \alpha = 1\) .

Q:

The bearings of towns \(Y\) and \(Z\) from town \(X\) are \(060 ^ { \circ } T\) and \(330 ^ { \circ } T\) respectively.

(a) Show that \(\angle Z X Y = 90 ^ { \circ }\) .

(b) Given that town \(Z\) is \(80 km\) from town \(X\) and that \(\angle X Y Z = 50 ^ { \circ }\) , find, correct to the nearest kilometre, how far town \(Y\) is from town \(X\) .

Q:

A golfer at \(G\) wishes to hit a shot between two trees \(P\) and \(Q ,\) as shown in the diagram opposite. The trees are \(31\) metres apart, and the golfer is \(74\) metres from \(P\) and \(88\) metres from \(P\) . Find the angle within which the golfer must play the shot, correct to the nearest degree.

Q:

Draw a sketch of, then find the equations of the sides of:

(a) the rectangle with vertices \(P ( 3 , - 7 ) , Q ( 0 , - 7 ) , R ( 0 , - 2 )\) and \(S ( 3 , - 2 )\) ,

(b) the triangle with vertices \(F ( 3,0 ) , G ( - 6,0 )\) and \(H ( 0,12 )\) .

Q:

From a ship sailing due north, a lighthouse is observed to be on a bearing of \(042 ^ { \circ } T\) . Later, when the ship is \(2\) nautical miles from the lighthouse, the bearing of the lighthouse from the ship is \(148 ^ { \circ } T\) . Find, correct to three significant figures, the distance of the lighthouse from the initial point of observation.

Q:

The three points \(A ( 1,0 ) , B ( 0,8 )\) and \(C ( 7,4 )\) form a triangle. Let \(\theta\) be the angle between \(A C\) and the \(x\) -axis.

(a) Find the gradient of the line \(A C\) and hence determine \(\theta\) , correct to the nearest degree.

(b) Derive the equation of \(A C\) .

(c) Find the coordinates of the midpoint \(D\) of \(A C\) .

(d) Show that \(A C\) is perpendicular to \(B D\) . (e) What type of triangle is \(A B C\) ? (f) Find the area of this triangle. (g) Write down the coordinates of a point \(E\) such that the parallelogram \(A B C E\) is a rhombus.

Q:

Find the points \(A\) and \(B\) where each line below meets the \(x\) -axis and \(y\) -axis respectively. Hence find the gradient of \(A B\) and its angle of inclination \(\alpha\) (correct to the nearest degree).

(a) \(y = 3 x + 6\)

(c) \(3 x + 4 y + 12 = 0\) (e) \(4 x - 5 y - 20 = 0\)

(b) \(y = - \frac { 1 } { 2 } x + 1\)

(d) \(\frac { x } { 3 } - \frac { y } { 2 } = 1\) (f) \(\frac { x } { 2 } + \frac { y } { 5 } = 1\)

Q:

Let the circle have centre the origin \(O\) and radius \(r\) . Let the diameter \(A B\) lie on the \(x\) -axis. Let \(A = ( r , 0 )\) and \(B = ( - r , 0 )\) . Let \(P ( a , b )\) be any point on the circle.

(a) Find \(P O ^ { 2 }\) and hence explain why \(a ^ { 2 } + b ^ { 2 } = r ^ { 2 }\) .

(b) Find the gradients of \(A P\) and \(B P\) .

(c) Show that the product of these gradients is \(\frac { b ^ { 2 } } { a ^ { 2 } - r ^ { 2 } }\) .

(d) Use parts (a) and (c) to show that \(A P \perp B P\) .

Q:

(a) Find the gradient of the interval \(A B\) , where \(A = ( 3,0 )\) and \(B = ( 5 , - 2 )\) .

(b) Find \(a\) if \(A P \perp A B\) , where \(P = ( a , 5 )\) .

(c) Find the point \(Q ( b , c )\) if \(B\) is the midpoint of \(A Q\)

(d) Find \(d\) if the interval \(A D\) has length \(5\) , where \(D = ( 6 , d )\)

Q:

(a) Find the midpoint of the interval joining \(A ( 4,9 )\) and \(C ( - 2,3 )\) .

(b) Find the midpoint of the interval joining \(B ( 0,4 )\) and \(D ( 2,8 )\)

(c) What can you conclude about the diagonals of the quadrilateral \(A B C D\) ?

(d) What sort of quadrilateral is \(A B C D ?\) [HINT: See Box \(4\) above.]

Q:

(a) The points \(A ( 1 , - 2 ) , B ( 5,6 )\) and \(C ( - 3,2 )\) are the vertices of a triangle, and \(P , Q\) and \(R\) are the midpoints of \(B C , C A\) and \(A B\) respectively. (i) Find the equations of the three medians \(B Q , C R\) and \(A P\) . (ii) Find the intersection of \(B Q\) and \(C R\) , and show that it lies on the third median \(A P\) .

(b) THEOREM: The three medians of a triangle are concurrent. (Their point of intersec- tion is called the centroid.) Prove that the theorem is true for any triangle by choosing as vertices \(A ( 6 a , 6 b\) ), \(B ( - 6 a , - 6 b )\) and \(C ( 0,6 c )\) , and following these steps. (i) Find the midpoints \(P , Q\) and \(R\) of \(B C , C A\) and \(A B\) respectively. (ii) Show that the median through \(C\) is \(x = 0\) and find the equations of the other two medians. (iii) Find the point where the median through \(C\) meets the median through \(A\) , and show that this point lies on the median through \(B\) .

Q:

In \(\triangle A B C , \angle B = 90 ^ { \circ }\) and \(\angle A = 31 ^ { \circ }\) . The point \(P\) lies on \(A B\) such that \(A P = 20 cm\) and \(\angle C P B = 68 ^ { \circ }\) .

(a) Explain why \(\angle A C P = 37 ^ { \circ }\) .

(b) Use the sine rule to show that \(P C = \frac { 20 \sin 31 ^ { \circ } } { \sin 37 ^ { \circ } }\) .

(c) Hence find \(P B\) , correct to the nearest centimetre.

Q:

Two towers \(A B\) and \(P Q\) stand on level ground. Tower \(A B\) is \(12\) metres taller than tower \(P Q\) . From \(A\) , the angles of depression of \(P\) and \(Q\) are \(28 ^ { \circ }\) and \(64 ^ { \circ }\) respectively.

(a) Use \(\triangle A K P\) to show that \(K P = B Q = 12 \tan 62 ^ { \circ }\) .

(b) Use \(\triangle A B Q\) to show that \(A B = 12 \tan 62 ^ { \circ } \tan 64 ^ { \circ }\) .

(c) Hence find the height of the shorter tower, correct to the nearest metre.

(d) Solve the problem again, by using \(\triangle A K P\) to find \(A P\) , and then using the sine rule in \(\triangle A P Q\) .

Q:

Let \(A B C\) be a triangle and let \(D\) be the midpoint of \(A C\) . Let \(B D = m\) and \(\angle A D B = \theta\) .

(a) Simplify \(\cos ( 180 ^ { \circ } - \theta )\) .

(b) Show that \(\cos \theta = \frac { 4 m ^ { 2 } + b ^ { 2 } - 4 c ^ { 2 } } { 4 m b }\) , and write downI a

(c) Hence show that \(a ^ { 2 } + c ^ { 2 } = 2 m ^ { 2 } + \frac { 1 } { 2 } b ^ { 2 }\) .

Q:

As was discussed in Section 3F, the circle with centre \(( h , k )\) and radius \(r\) has equation \(( x - h ) ^ { 2 } + ( y - k ) ^ { 2 } = r ^ { 2 }\) . By identifying the centre and radius, find the equations of:

(a) the circle with centre \(( 5 , - 2 )\) and passing through \(( - 1,1 )\) ,

(b) the circle with \(K ( 5,7 )\) and \(L ( - 9 , - 3 )\) as endpoints of a diameter.

Q:

Use gradients to show that each quadrilateral \(A B C D\) below is a parallelogram. Then use the definitions in Box \(3\) of the notes to show that it is:

(a) a rhombus, for the vertices \(A ( 2,1 ) , B ( - 1,3 ) , C ( 1,0 )\) and \(D ( 4 , - 2 )\) ,

(b) a rectangle, for the vertices \(A ( 4,0 ) , B ( - 2,3 ) , C ( - 3,1 )\) and \(D ( 3 , - 2 )\)

(c) a square, for the vertices \(A ( 3,3 ) , B ( - 1,2 ) , C ( 0 , - 2 )\) and \(D ( 4 , - 1 )\)

Q:

A vertical pole stands on level ground. From a point on the ground \(8\) metres from its base, the angle of elevation of the top of the pole is \(38 ^ { \circ }\) . Find the height of the pole, correct to the nearest centimetre.

Q:

The quadrilateral \(A B C D\) has vertices \(A ( - 1,1 ) , B ( 3 , - 1 ) , C ( 5,3 )\) and \(D ( 1,5 )\) . Use the definitions of the special quadrilaterals in Box \(3\) above to answer these questions.

(a) Show that the opposite sides are parallel, and hence that \(A B C D\) is a parallelogram.

(b) Show that \(A B \perp B C\) , and hence that \(A B C D\) is a rectangle.

(c) Show that \(A B = B C\) , and hence that \(A B C D\) is a square.

Q:

Draw a separate sketch of \(\triangle A B C\) for each part. In your answers, give lengths and areas correct to four significant figures, and angles correct to the nearest minute.

(a) Find \(c\) , given that \(a = 12 cm , b = 14 cm\) and \(\angle C = 35 ^ { \circ }\) .

(b) Find \(b\) , given that \(a = 12 cm , \angle A = 47 ^ { \circ }\) and \(\angle B = 80 ^ { \circ }\) .

(c) Find \(\angle B\) , given that \(a = 12 cm , b = 24\) and \(\angle A = 23 ^ { \circ }\) .

(d) Find \(\angle A\) , given that \(a = 12 cm , b = 8\) and \(c = 11\) . (e) Find the area, given that \(a = 12 cm , c = 9\) and \(\angle B = 28 ^ { \circ }\) . (f) Find \(\angle C\) , given that \(a = 12 cm , b = 7\) and the area is \(33 cm ^ { 2 }\) . (g) Find \(c\) , given that \(a = 12 cm , \angle B = 65 ^ { \circ }\) and the area is \(60 cm ^ { 2 }\) .

Q:

The line \(\ell\) crosses the \(x\) - and \(y\) -axes at \(L ( - 4,0 )\) and \(M ( 0,3 )\) . The point \(N\) lies on \(\ell\) and \(P\) is the point \(P ( 0,8 )\) .

(a) Copy the sketch and find the equation of \(\ell\) .

(b) Find the lengths of \(M L\) and \(M P\) and hence show that

(c) If \(M\) is the midpoint of \(L N\) , find the coordinates of \(N\) .

(d) Show that \(L N P L = 90 ^ { \circ }\) . (e) Write down the equation of the circle through \(N , P\) and \(L\) .

Q:

Place three vertices of the parallelogram at \(A ( 0,0 ) , B ( 2 a , 2 b )\) and \(D ( 2 c , 2 d )\) .

(a) Use gradients to show that with \(C = ( 2 a + 2 c , 2 b + 2 d )\) , the quadrilateral \(A B C D\) is a parallelogram.

(b) Find the midpoints of the diagonals \(A C\) and \(B D\) .

(c) Explain why this proves that the diagonals bisect each other.

Q:

The quadrilateral \(A B C D\) has vertices at the points \(A ( 1,0 ) , B ( 3,1 ) , C ( 4,3 )\) and \(D ( 2,2 )\) . [HINT: You should look at Boxes \(3\) and \(4\) in the notes above to answer this question.]

(a) Show that the intervals \(A C\) and \(B D\) bisect each other, by finding the midpoint of each and showing that these midpoints coincide.

(b) What can you conclude from part (a) about what type of quadrilateral \(A B C D\) is?

(c) Show that \(A B = A D\) . What can you now conclude about the quadrilateral \(A B C D\) ?

Q:

Sketch \(\triangle A B C\) in which \(a = 2.8 cm , b = 2.7 cm\) and \(A = 52 ^ { \circ } 21 ^ { \prime }\) .

(a) Find \(B\) , correct to the nearest minute.

(b) Hence find \(C\) , correct to the nearest minute.

(c) Hence find the area of \(\triangle A B C\) in \(cm ^ { 2 }\) , correct to two decimal places.

Q:

A quadrilateral has vertices \(W ( 2,3 ) , X ( - 7,5 ) , Y ( - 1 , - 3 )\) and \(Z ( 5 , - 1 )\) .

(a) Show that \(W Z\) is parallel to \(X Y ,\) but that \(W Z\) and \(X Y\) have different lengths.

(b) What type of quadrilateral is \(W X Y Z\) ? [HinT: Look at Boxes \(3\) and \(4\) above.]

(c) Show that the diagonals \(W Y\) and \(X Z\) are perpendicular.

Q:

The lines \(2 x + y - 5 = 0\) and \(x - y + 2 = 0\) intersect at \(A\) .

(a) Write down the general equation of a line through \(A\) , and show that it can be written in the form \(x ( 2 + k ) + y ( 1 - k ) + ( 2 k - 5 ) = 0\) .

(b) Find the value of \(k\) that makes the coefficient of \(x\) zero, and hence find the equation of the horizontal line through \(A\) .

(c) Find the value of \(k\) that makes the coefficient of \(y\) zero, and hence find the equation of the vertical line through \(A\) .

(d) Hence write down the coordinates of \(A\) .

Q:

(a) On a number plane, plot the points \(A ( 4,3 ) , B ( 0 , - 3 )\) and \(C ( 4,0 )\) .

(b) Find the equation of \(B C\) .

(c) Explain why \(O A B C\) is a parallelogram.

(d) Find the area of \(O A B C\) and the length of the diagonal \(A B\) .

Q:

(a) Write down the centre and radius of the circle with equa- tion \(( x + 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 4\) . Then find the distance from the line \(2 y - x + 8 = 0\) to the centre.

(b) Use Pythagoras' theorem to determine the length of the chord cut off from the line by the circle.

Q:

A motorist drove \(70 km\) from town \(A\) to town \(B\) on a bearing of \(056 ^ { \circ } T\) , and then drove \(90 km\) from town \(B\) to town \(C\) on a bearing of \(146 ^ { \circ } T\) .

(a) Explain why \(\angle A B C = 90 ^ { \circ }\) .

(b) How far apart are the towns \(A\) and \(C\) , correct to the nearest kilometre?

(c) Find \(\angle B A C\) , and hence find the bearing of town \(C\) from town \(A\) , correct to the nearest degree.

Q:

(a) Find the point \(P\) of intersection of \(x + y - 2 = 0\) and \(2 x - y - 1 = 0\) .

(b) Show that \(P\) satisfies the equation \(x + y - 2 + k ( 2 x - y - 1 ) = 0\) .

(c) Find the equation of the line through \(P\) and \(Q ( - 2,2 ) :\) (i) using the coordinates of both \(P\) and \(Q\) , (ii) without using the coordinates of \(P\) . The two answers should be the same.

Q:

Put the equation of each line in gradient-intercept form and hence write down the gradient. Then find, in gradient-intercept form, the equation of the line that is: (i) parallel to it through \(A ( 3 , - 1 )\) , (ii) perpendicular to it through \(B ( - 2,5 )\) .

(a) \(2 x + y + 3 = 0\)

(b) \(5 x - 2 y - 1 = 0\)

(c) \(4 x + 3 y - 5 = 0\)

Q:

A triangle has vertices \(R ( 3 , - 4 ) , S ( - 6,1 )\) and \(T ( - 2 , - 2 )\) .

(a) Find the gradient of the line \(S T\) , and hence find its equation.

(b) Find the perpendicular distance from \(R\) to \(S T\) .

(c) Find the length of the interval \(S T\) and hence find the area of \(\triangle R S T\) .

Q:

(a) Find the perpendicular distance from \(A ( - 2 , - 3 )\) to the line \(5 x + y + 2 = 0\) .

(b) Find the distance between the parallel lines \(3 x + y - 3 = 0\) and \(3 x + y + 7 = 0\) . [HiNT: Choose a point on one line and find its perpendicular distance from the other.]

(c) Find \(k\) if \(\ell : 3 x + 4 y + 3 = 0\) is a tangent to the circle with centre \(A ( k , - 5 )\) and radius \(3\) .

Q:

(a) Find the distance of each point \(A ( 1,4 ) , B ( 2 , \sqrt { 13 } ) , C ( 3,2 \sqrt { 2 } )\) and \(D ( 4,1 )\) from the origin \(O\) . Hence explain why the four points lie on a circle with centre the origin.

(b) What are the radius, diameter, circumference and area of this circle?

Q:

Two triangles are shown, with sides \(6 cm\) and \(4 cm\) , in which the angle opposite the \(4 cm\) side is \(36 ^ { \circ }\) . Find, in each case, the angle opposite the \(6 cm\) side, correct to the nearest degree.

Q:

Town \(A\) is \(23 km\) from landmark \(L\) in the direction \(N 56 ^ { \circ } W\) , and town \(B\) is \(31 km\) from \(L\) in the direction \(N 46 ^ { \circ } E\) .

(a) Find how far town \(B\) is from town \(A\) . (Answer correct to the nearest \(km\) .)

(b) Find the bearing of town \(B\) from town \(A\) . (Answer correct to the nearest degree.)

Q:

The quadrilateral \(A B C D\) has vertices \(A ( 1 , - 4 ) , B ( 3,2 ) , C ( - 5,6 )\) and \(D ( - 1 , - 2 )\) .

(a) Find the midpoints \(P\) of \(A B , Q\) of \(B C , R\) of \(C D\) , and \(S\) of \(D A\) .

(b) Prove that \(P Q R S\) is a parallelogram by showing that \(P Q \| R S\) and \(P S \| Q R\) .

Q:

The vertices of a triangle are \(P ( - 1,0 )\) and \(Q ( 1,4 )\) and \(R\) , where \(R\) lies on the \(x\) -axis and \(\angle Q P R = \angle Q R P = \theta\) .

(a) Find the coordinates of the midpoint of \(P Q\) .

(b) Find the gradient of \(P Q\) and show that \(\tan \theta = 2\) .

(c) Show that \(P Q\) has equation \(y = 2 x + 2\) .

(d) Explain why \(Q R\) has gradient \(- 2\) , and hence find its equation. (e) Find the coordinates of \(R\) and hence the area of triangle \(P Q R\) . (f) Find the length \(Q R\) , and hence find the perpendicular distance from \(P\) to \(Q R\) .

Q:

From the ends of a straight horizontal road \(1 km\) long, a balloon directly above the road is observed to have angles of elevation of \(57 ^ { \circ }\) and \(33 ^ { \circ }\) respectively. Find, correct to the nearest metre, the height of the balloon above the road.

Q:

The diagram to the right shows an isosceles triangle in which the apex angle is \(35 ^ { \circ }\) . Its area is \(35 cm ^ { 2 }\) . Find the length of the equal sides, correct to the nearest millimetre.

Q:

There are two triangles that have sides \(9 cm\) and \(5 cm\) , and in which the angle op- posite the \(5 cm\) side is \(22 ^ { \circ }\) . Find, in each case, the size of the angle opposite the \(9 cm\) side, correct to the nearest degree.

Q:

Similarly, show that each triangle below is right-angled. Then find the lengths of the sides enclosing the right angle, and calculate the area of each triangle.

(a) \(P ( 2 , - 1 ) , Q ( 3,3 ) , R ( - 1,4 )\)

(b) \(X ( - 1 , - 3 ) , Y ( 2,4 ) , Z ( - 3,2 )\)

Q:

Draw a sketch of the quadrilateral with vertices \(J ( 2 , - 5 ) , K ( - 4,3 ) , L ( 4,9 )\) and \(M ( 13 , - 3 )\) .

(a) Show that \(J K L M\) is a trapezium with \(J K \| M L\) .

(b) Find the lengths of the parallel sides \(J K\) and \(M L\) .

(c) Find the equation of the line \(J K\) .

(d) Find the perpendicular distance from \(M\) to \(J K\) . (e) Hence find the area of the trapezium \(J K L M\) .

Q:

Substitute \(y = 0\) and \(x = 0\) into the equation of each line below to find the points \(A\) and \(B\) where the line crosses the \(x\) -axis and \(y\) -axis respectively. Hence sketch the line.

(a) \(5 x + 3 y - 15 = 0\)

(b) \(2 x - y + 6 = 0\)

(c) \(3 x - 5 y + 12 = 0\)

Q:

Consider the line \(\ell : k x + 3 y + 12 = 0\) , where \(k\) is a constant.

(a) Find \(k\) if \(\ell\) passes through \(( 2,4 ) .\)

(d) Find \(k\) if \(\ell\) is parallel to \(6 x - y - 7 = 0\)

(b) Find \(k\) if \(\ell\) has \(x\) -intercept \(- 36 .\) (e) Find \(k\) if \(\ell\) is perpendicular to \(6 x - y - 7 = 0\)

(c) Find \(k\) if \(\ell\) has gradient \(9 .\) (f) Explain why \(\ell\) can never have \(y\) -intercept \(1\)

Q:

The points \(A ( 3,1 ) , B ( 10,2 ) , C ( 5,7 )\) and \(D ( - 2,6 )\) are the vertices of a quadrilateral.

(a) Find the lengths of all four sides.

(b) What sort of quadrilateral is \(A B C D\) ? [HINT: See Box \(4\) above.]

Q:

The points \(A ( 1,4 ) , B ( 5,0 )\) and \(C ( 9,8 )\) form the vertices of a triangle.

(a) Find the coordinates of the midpoints \(P\) and \(Q\) of \(A B\) and \(A C\) respectively.

(b) Show that \(P Q\) is parallel to \(B C\) and half its length.

Q:

(a) Find the side lengths of the triangle with vertices \(X ( 0 , - 4 ) , Y ( 4,2 )\) and \(Z ( - 2,6 )\) .

(b) Show that \(\triangle X Y Z\) is a right-angled isosceles triangle by showing that its side lengths satisfy Pythagoras' theorem.

(c) Hence find the area of \(\triangle X Y Z\) .

Q:

Find the gradient of each line below. Hence find, in gradient-intercept form, the equation of a line passing through \(A ( 0,3 )\) and:

(a) \(2 x + y + 3 = 0\)

(b) \(5 x - 2 y - 1 = 0\)

(c) \(3 x + 4 y - 5 = 0\)

Q:

Draw on a number plane the triangle \(A B C\) with vertices \(A ( 5,0 ) , B ( 8,4 )\) and \(C ( 0,10 )\)

(a) Show that the line \(A B\) has equation \(3 y = 4 x - 20\) .

(b) Show that the gradient of \(B C\) is \(- \frac { 3 } { 4 }\) .

(c) Hence show that \(A B\) and \(B C\) are perpendicular.

(d) Show that the interval \(A B\) has length \(5\) units. (e) Show that the triangles \(A O C\) and \(A B C\) are congruent. (f) Find the area of quadrilateral \(O A B C\) . (g) Find the distance from \(D ( 0,8 )\) to the line \(A B\) .

Q:

(a) Show that the points \(A ( - 5,0 ) , B ( 5,0 )\) and \(C ( 3,4 )\) all lie on the circle \(x ^ { 2 } + y ^ { 2 } = 25\) .

(b) Explain why \(A B\) is a diameter of the circle.

(c) Show that \(A C \perp B C\) .

Q:

Write down the gradient \(m\) of each line. Then use the formula gradient \(= \tan \alpha\) to find its angle of inclination \(\alpha\) , correct to the nearest minute where appropriate.

(a) \(y = x + 3\)

(b) \(y = - x - 16\)

(c) \(y = 2 x\)

(d) \(y = - \frac { 3 } { 4 } x\)

Q:

Find the range of values that \(k\) may take if:

(a) the line \(y - x + k = 0\) is more than \(\frac { 1 } { \sqrt { 2 } }\) units from the point \(( 2,7 )\)

(b) the line \(x + 2 y - 5 = 0\) is at most \(\sqrt { 5 }\) units from the point \(( k , 3 )\) .

Q:

A ladder of length \(5\) metres is placed on level ground against a vertical wall. If the foot of the ladder is \(1.5\) metres from the base of the wall, find, correct to the nearest degree, the angle at which the ladder is inclined to the ground.

Q:

(a) It is known that the line \(\ell : x + 2 y + 10 = 0\) is tangent to the circle \(C : x ^ { 2 } + y ^ { 2 } = 20\) at the point \(T\) . Use the fact that a tangent is perpendicular to the radius at the point of contact to write down the equation of the radius \(O T\) of the circle.

(b) Without actually finding the coordinates of \(T\) , use the result of part (a) to find the equation of the line through \(S ( 1 , - 3 )\) and the point of contact \(T\) .

Q:

Write down the centre and radius of each circle. Then use the perpendicular distance formula to determine how many times the given line intersects the circle.

(a) \(3 x - 5 y + 16 = 0 , x ^ { 2 } + y ^ { 2 } = 5\)

(c) \(3 x - y - 8 = 0 , ( x - 1 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 10\)

(b) \(7 x + y - 10 = 0 , x ^ { 2 } + y ^ { 2 } = 2\)

(d) \(x + 2 y + 3 = 0 , ( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 6\)

Q:

(a) Show that the three lines \(\ell _ { 1 } : 2 x - 3 y + 13 = 0 , \ell _ { 2 } : x + y - 1 = 0\) and \(\ell _ { 3 } : 4 x + 3 y - 1 = 0\) are concurrent by the following method. (i) Without finding any points of intersection, find the equation of the line that passes through the intersection of \(\ell _ { 1 }\) and \(\ell _ { 2 }\) and is parallel to \(\ell _ { 3 }\) . (ii) Show that this line is the same line as \(\ell _ { 3 }\) .

(b) Use the same method as in the previous question to test each family of lines for concurrency. (i) \(\ell _ { 1 } : 2 x - y = 0 , \ell _ { 2 } : x + y = 9\) and \(\ell _ { 3 } : x - 3 y + 15 = 0\) (ii) \(\ell _ { 1 } : x + 4 y + 6 = 0 , \ell _ { 2 } : x + y - 3 = 0\) and \(\ell _ { 3 } : 7 x - 3 y - 10 = 0\)

Q:

Let \(M\) be the point of intersection of the lines \(\ell _ { 1 } : 3 x - 4 y - 5 = 0\) and \(\ell _ { 2 } : 4 x + y + 7 = 0\) . Write down the equation of the general line through \(M\) . Hence, without actually finding the coordinates of \(M\) , find the equation of:

(a) the line through \(M\) and \(A ( 1,1 ) ,\)

(c) the vertical line through \(M\)

(b) the line through \(M\) with gradient \(- 3 ,\)

(d) the horizontal line through \(M\)

Q:

Find the gradients of the four lines in each part. Hence state what sort of special quadri- lateral they enclose.

(a) \(3 x + y + 7 = 0 , x - 2 y - 1 = 0 , 3 x + y + 11 = 0 , x - 2 y + 12 = 0\)

(b) \(4 x - 3 y + 10 = 0 , 3 x + 4 y + 7 = 0 , 4 x - 3 y - 7 = 0 , 3 x + 4 y + 1 = 0\)

Q:

There are three landmarks, \(P , Q\) and \(R\) . It is known that \(R\) is \(8.7 km\) from \(P\) and \(9.3 km\) from \(Q\) , and that \(\angle P R Q = 79 ^ { \circ } 32 ^ { \prime }\) . Draw a diagram and find the distance between \(P\) and \(Q\) , in kilometres correct to one decimal place.

Q:

The radius of a cylinder is multipled by \( 6 \) while the height is kept the same. What effect does this have on the volume of the cylinder? 

The volume will be multipled by - 

A. \( 6 \)                     C. \( 36 \) 

B. \( 12 \)                   D. \( 216\) 

Q:

A square pyramid measuring \( 10 \) yd along each edge of the base with a height of \( 6 \) yd. Find the volume of the square pyramid. 

Q:

The three-dimensional figure below is a solid rectangular prism with a hole in the shape of another rectangular prism going through the center of it. Find the volume of the solid cubic centimeters.  

 

Q:

Coordinates for a polygon are \( ( 1,4 ) , ( 6,4 ) , \) and \( ( 6,1 ) \) . What type of polygon is formed by these points? (hint: graph the figure and then apply properties) 

A. Isosceles Triangle 

B. Equilateral Triangle 

C. Right Triangle 

D. Trapezoid 

Q:

What are the side lengths of the rectangle? 

Area \( = 40 \) in 

Perimeter \( = 26 \) in 

Q:

A cereal box is a rectangular prism. If the area of the base is doubled and the height is not changed, how many times greater will the volume be? 

Q:

A square has a perimeter of \( 36 \) units. 

One vertex of the square is located at \( ( 3,5 ) \) on the coordinate grid. 

What could be the \( x \) - and \( y \) -coordinates of another vertex of the square? 

Q:

Compute the area of a circle with a diameter of \( 15 \) inches. You do not need to include the unit measurement or unit type. Only the numerical answer is necessary. Round to the nearest tenth if necessary. Use the value of \( 3.14 \) for pi. 

Q:

What is the lateral surface area (the area not including the circular bases) of a cylinder whose base diameter is \( 6 cm \) and whose height is \( 9 cm \) ? Round your answer to the nearest whole number. 

Q:

Use the given information to prove that \(\triangle PQR\cong \triangle TSR\)

\(Given: QR\cong SR \\\ \ \ \ \ \ \ \ \ \ \ \ \angle PQR\cong \angle TSR\)

\(Prove: \triangle PQR\cong \triangle TSR\)

Q:

The ratio of the lengths of the corresponding sides of two similar rectangles is \( 12 : 5 \) . What is the ratio of the areas between the larger and smaller rectangles? (Write your answer in the form a:b) 

Q:

The ratio of the lengths of corresponding sides of two similar triangles is 3:5. The larger triangle has an area of \( 100 \) square inches. What is the area of the smaller triangle? 

Q:

a. The diameter of Circle \( J \) is \( 18 cm \) . The diameter of Circle K is \( 31 cm \) . Which circle has the greater area and by how much? Use \( 3.14 \) for \( \pi \) . 

b. Describe how you can compare the areas of two circles. 

Q:

You and your brother Billy bought walkie talkies to use outside. The range of the walkie talkies only covers a circular area with a radius of \( 30 \) feet. How many feet of coverage will the walkie talkies get before you lose contact?