Question

Erin says the first step is to set Latex: 2x+72x+7equal to Latex: 5x-2\textsf{.}Antonio says the first step is to set Latex: 2x+72x+7plus Latex: 5x-25x−2equal to Latex: 180\textsf{.}Choose either Erin's or Antonio's statement and explain why it is correct or incorrect. If the statement is correct, explain how you know it's correct. If the statement is incorrect, explain why it's wrong.If you were to solve this problem, which steps would you take? Include any theorems, definitions, or reasons that explain the steps. Make sure you include all steps needed to solve for Latex: m\angle A\textsf{.}Please number your responses to the questions as they are shown (1 and 2).

Answer 1

1)

Point A is the center of the circle

DE is an arc

DC and CE are tangents to the circle.

Erin is incorrect because there is no theorem stating that the angle formed by two tangents to a circle is equal to the arc intercepted by the tangents.

Antonio is incorrect because there is no theorem stating that the sum of the angle formed by two tangents to a circle and the arc intercepted by the tangents is 180.

2) The first step is to apply the theorem which states that angle formed outside a circle by the intersection of two tangents to the circle is equal to the half of the difference between the measure of the intercepted arc. From the triangle, the intercepted arcs are

5x - 2 and

360 - (5x - 2) because the measure of the circumference of a circle is 360.

The angle formed by the tangents is 2x + 7. By applying the theorem, we have

2x + 7 = 1/2(360 - (5x - 2) - (5x - 2))

2x + 7 = 1/2(360 - 5x + 2 - 5x + 2))

2x + 7 = 1/2(360 + 2 + 2 - 5x - 5x)

2x + 7 = 1/2(364 - 10x)

By crossmultiplying, we have

2(2x + 7) = 1/2 * 2(364 - 10x)

4x + 14 = 364 - 10x

Adding 10x to both sides of the equation, we have

4x + 10x + 14 = 364 - 10x + 10x

14x + 14 = 364

Subtracting 14 from both sides of the equation,

14x + 14 - 14 = 364 - 14

14x = 350

Dividing both sides of the equation by 14, we have

14x/14 = 350/14

x = 25

The next step is to apply the central angle theorem which states that the angle formed by two radii with the vertex at the center of the circle is equal to the arc intercepted by the two radii. DA and AE are two radii and the angle formed is A.The arc intercepted is DE. By applying the central angle theorem,

Angle A = arc DE = 5x - 2

Substituting x = 25 into angle A = 5x - 2, we have

Angle A = 5 * 25 - 2 = 123

Angle A = 123 degrees