Question

a theorem in geometry states that the measure of an inscribed angle is half the measure of its intercepted arc. in the figure, ∠c intercepts arc ab and (overline{ab}) is the diameter of the circle. which equation is a step in showing that (mangle c = 90^{circ})?

Explanation:

Step1: Recall slope - formula

The slope of line $AC$ with $A(-c,0)$ and $C(a,b)$ is $m_{AC}=\frac{b - 0}{a-(-c)}=\frac{b}{a + c}$. The slope of line $BC$ with $B(c,0)$ and $C(a,b)$ is $m_{BC}=\frac{b-0}{a - c}=\frac{b}{a - c}$.

Step2: Recall perpendicular - lines property

Two lines are perpendicular if the product of their slopes is $- 1$. Since $\angle C = 90^{\circ}$, lines $AC$ and $BC$ are perpendicular, so $m_{AC}\times m_{BC}=-1$. Substituting the slopes, we get $(\frac{b}{a - c})(\frac{b}{a + c})=-1$.

Answer:

$(\frac{b}{a - c})(\frac{b}{a + c})=-1$