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 ab is the diameter of the circle. which equation is a step in showing that m∠c = 90°? a(-c,0) b(c,0) c(a,b) (\\(\frac{b}{a - c}\\))(\\(\frac{b}{a + c}\\))=-1 (\\(\frac{b}{-c}\\))(\\(\frac{b}{c}\\))=-1 (\\(\frac{b}{c - a}\\))(\\(\frac{b}{c + a}\\))=-1 (\\(\frac{b}{a - c}\\))(\\(\frac{b}{a + c}\\))=1

Explanation:

Step1: Recall slope - formula

The slope of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. 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: Use perpendicular - slope property

If two lines are perpendicular, the product of their slopes is $- 1$. Since $\angle C = 90^{\circ}$, lines $AC$ and $BC$ are perpendicular. So, $m_{AC}\times m_{BC}=-1$, which gives $(\frac{b}{a - c})(\frac{b}{a + c})=-1$.

Answer:

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