Chapter Test
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By the Inscribed Right Triangle Theorem, a right triangle is inscribed in a circle if and only if its hypotenuse is a diameter of the circle. Also, the Tangent Line to Circle Theorem tells that a line is tangent to the circle if and only if it is perpendicular to the radius drawn to the point of tangency.
m∠1=90^(∘), m∠2=90^(∘)
Consider the given diagram.
We will find the measures of ∠1 and ∠2 one at a time.
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Inscribed Right Triangle Theorem |
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If one side of an inscribed triangle is the diameter of the circle, then the angle opposite the diameter is the right angle. |
With this in mind, we can find the measure of ∠1.
By the theorem, the measure of ∠1 equals 90^(∘). m∠1=90^(∘)
Note that the chord containing P, which is the center of the circle, is the diameter. Also, ∠2 represents the angle between the tangent ray and the diameter at the point of tangency as shown on the given diagram. In this case, we can consider the Tangent Line to Circle Theorem.
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Tangent Line to Circle Theorem |
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A line is tangent to the circle if and only if it is perpendicular to the radius drawn to the point of tangency. |
With this in mind, we can find the measure of ∠2.
By the theorem, the measure of ∠2 is expressed as follows. m∠2=90^(∘)