Question

HELPP 35 POINTS!!!!! Point Q is the center of Circle Q in the diagram below. The measure of angle XAY is 72° (m∠XAY = 72°). Since the measure of the intercepted arc XY is 144°, a student named Naomi mistakenly claims that the measure of ∠XBY is also 144°.
a. In a short paragraph, explain the source of Naomi’s error. In your response, be sure to describe the problem in terms of the following two formulas:

inscribed angle = 1/2 · (intercepted arc) and
intercepted arc = central angle.

Be sure to use important vocabulary like “inscribed angle,” “central angle”, and “intercepted arc”.

b. Make an estimate of the degree measure of ∠XBY. Justify your answer.

Answer 1

Answer:

Se explanation

Step-by-step explanation:

The diagram shows the circle with center Q. In this circle, angle XAY is inscribed angle subtended on the arc XY. Angle XQY is the central angle subtended on the same arc XY.

The inscribed angle theorem states that an angle inscribed in a circle is half of the central angle that subtends the same arc on the circle. Therefore,

$m\angle XBY=2m\angle XAY=2\cdot 72^{\circ}=144^{\circ}$

The measure of the intercepted arc XY is the measure of the central angle XQY and is equal to 144°.

All angles that have the same endpoints X and Y and vertex lying in the middle of the quadrilateral XAYQ have measures satisfying the condition

$72^{\circ}$

because angle XAY is the smallest possible angle subtended on the arc XY in the circle and angle XQY is the largest possible angle in the circle subtended on the arc XY.