Power of a Point Theorem

In math, power of point theorem is a theorem related to circle. Power of point theorem tells the relationship between the intersecting lines of a circle. Base on the intersecting lines, we have three possibilities for power of point theorem. This article gives a clear explanation of power of point theorem with some example problems.

Explanation to Power of Point Theorem;

Three possibilities for power of point theorem:
Case: 1



In the above figure, there are two intersecting lines intersect the circle insides. Then the power of point theorem is AE . CE = BE. DE

Case 2:



In the above figure, one of the line is tangent to the given circle. Then the power of point theorem is, AB2 = BC . BD

Case 3:



In the above figure, there are two lines intersect outside of the circle. Then the power of point theorem is CB . CA = CD . CE

Special case of power of point theorem:



In the above figure, there are two tangent lines. Then the power of point theorem is PA = PC

Example Problems to Power of Point Theorem:

Example: 1

Determine the unknown value of the following figure using power of point theorem.



Solution:

Given:

CB = 2

AB = 4

CD = x

DE = 1

Step 1:

Here two lines are intersecting at outside. So, the power of point theorem is, CB . CA = CD . CE

Step 2:

CB . CA = CB . (CB + CA)

= 2 . (2 + 4)

=2 . 6

= 12

Step 2:

CD . CE = x . (CD + DE)

= x . (x + 1)

= x2 + x

Step 3:

CB . CA = CD . CE

12 = x2 + x

x2 + x - 12 = 0

(x + 4)(x - 3) = 0

x = -4, 3 (Discard the negative answer)

x = 3

Answer: 1

Example: 2

Determine the unknown value of the following figure using power of point theorem.



Solution:

Given:

EA = x

ED = 2

EC = 5

EB = 5

Step 1:

Here two lines are intersecting at inside. So, the power of point theorem is,AE . CE = BE. DE

Step 2:

x . 5 = 5 . 2

5x = 10

x = `10/5`

x = 2

Answer: x = 2

Example Problems to Power of Point Theorem:

Problem: 1

Determine the unknown value of the following figure using power of point theorem.



Answer: 4

Problem: 2

Determine the unknown value of the following figure using power of point theorem.



Answer: 6