Question

Examine the diagram, where AB←→
and CD←→
are secants to the circle, and they intersect inside of the circle at point P,
which is not the center.

A circle, with no center shown, as described in the text. Line A B is secant to the circle at A and B, line C D is secant to the circle at C and D, and they intersect at point P. Segment A P equals 21, segment B P equals 2x, segment C P equals x plus 5, and segment D P equals 27.What is the value of x?
Responses

11
11

6
6

5
5

9

Answer 1

x = 11

Solution:
Since segments AB and CD are secants to the circle that intersect inside the circle at point P, we can use the Intercepted Arc Theorem. This theorem states that if two secants intersect inside a circle, then the measure of an angle formed is equal to half the sum of the intercepted arcs.

Let's denote the measure of arc ADB as m. Therefore, the measure of arc ACB is 2m.
Since segment AP equals 21 and segment DP equals 27, we can say that the sum of the intercepted arcs is equal to the measure of the entire circle, which is 360 degrees.

m + 21 + 27 + 2m = 360
3m + 48 = 360
3m = 312
m = 104

Now that we know the measure of arc ADB, we can find the value of x:
x + 5 + 104 = 360
x = 360 - 109
x = 251

Therefore, x = 11.