Power of a Point Theorem Calculators
Knowledge
What is the power of a point theorem?
The power of a point theorem is a summary theorem that unifies and generalizes the chord theorem, the tangent-secant theorem and the intersecting secants theorem. The following are the definitions of these three types of theorems.
(1) The chord theorem: for two intersecting chords in a circle (AC and BD), the product of the lengths of the two line segments divided by the intersection (E) is equal.
(2) The tangent-secant theorem: draw a tangent segment (AB) and a secant segment (BD) from the outside of the circle, and the square of this tangent segment is equal to the product of the secant segment (BC) and its outer secant segment (BD).
(3) The intersecting secants theorem: if two secant segments (CA and CE) are drawn to the circle from an outer point (C), then the product of one secant segment (CB) and its outer secant segment (CA) is equal to the product of the measure of the other secant segment (CD) and its outer secant segment (CE).
CB · CA = CD · CE
CA = CB + BA
CE = CD + DE
From the above theorem, it can be seen that drawing two lines from a point inside or outside a circle has a similar conclusion, regardless of whether they are positioned inside or outside the circle. This is the power of the point theorem.
The power of the point theorem practice problems
Now, let’s practice!
Question 1:
A circle intersects two lines at the points A, B, C and D. The same two lines meet at a point P, outside the circle. Given that PA=4, PB=8, PC=12, what's the length of PD?
Solution: using the power of the point formula, that gives you
PD = 3212 = 83
Question 2:
Find the value of x in the following diagram:
AB · AC = AD · AE
3 · (3+5) = x(x+10)
x2 + 10x - 24 = 0(x+10)
(x+12)(x-2) = 0(x+10)
x = 2(x+10)
Question 3:
Find the value of x in the following diagram:
EA · DA = CA · BA
6 · x = 4 · 1111111
x = 23111111
