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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?