A theorem in geometry states that the measure of an inscribed angle is half the measure of its intercepted arc. In the figure, ZC intercepts arc AB, and AB is the diameter of the circle. Which equation helps show that m‚à†ZC = 90¬∞?

Mathematics Questions

A theorem in geometry states that the measure of an inscribed angle is half the measure of its intercepted arc. In the figure, ZC intercepts arc AB, and AB is the diameter of the circle. Which equation helps show that m‚à†ZC = 90¬∞?

Short Answer

The slope of line AC is calculated as m1 = (a + c) / b, while the slope of line CB is m2 = (c – a) / -b. For the lines to be perpendicular, the product of their slopes must equal -1, confirming their perpendicularity if the equation m1 * m2 = -1 holds true.

Step-by-Step Solution

Step 1: Evaluate the Slope of Line AC

To find the slope of line AC, denote the coordinates involved. The slope can be calculated using the formula: m₁ = (y₂ – y₁) / (x₂ – x₁). By substituting the points, we find:

m₁ = a – (-c) / (b – 0)
m₁ = (a + c) / b

Step 2: Evaluate the Slope of Line CB

Now we’ll determine the slope of line CB using the same method. Again, we substitute the coordinates of points C and B into the slope formula:

m₂ = (y₂ – y₁) / (x₂ – x₁)
m₂ = (c – a) / (0 – b)
m₂ = (c – a) / -b

Step 3: Establishing Perpendicular Lines

For the lines to be perpendicular, the product of their slopes (m₁ and m₂) must yield -1. Hence, we equate:

m₁ * m₂ = -1
[(a + c) / b] * [(c – a) / -b] = -1

This leads us to the conclusion that these two lines are indeed perpendicular if the equation holds true.

A theorem in geometry states that…