Question

A circle is shown. Secant A D and tangent E D intersect at point D outside of the circle. Secant A D intersects the circle at point B. The length of A B is a, the length of B D is 10, and the length of D E is 12.

Which equation results from applying the secant and tangent segment theorem to the figure?

Answer 1

By applying the secant and tangent segment theorem to the figure, we can set up the following equation:

AD * AE = AB^2

We know that AB = a, and we can find AD and AE by using the segment addition postulate:

AD = AB + BD = a + 10

AE = DE = 12

Substituting these values into the equation, we get:

(a + 10) * 12 = a^2

Expanding the left side, we get:

12a + 120 = a^2

Rearranging and simplifying, we get:

a^2 - 12a - 120 = 0

This is the equation that results from applying the secant and tangent segment theorem to the figure.