Answer:
7) m∠SPD = 12.5°
8) m∠RQT = 90°
9) m∠SRT = 135°
10) m∠DPQ = 22.5°
11) m∠RTP = 135°
Step-by-step explanation:
Question 7
To find the measure of angle SPD, we can use the Intersecting Secants Theorem.
According to the theorem, if two secant segments are drawn to the circle from one exterior point, the measure of the angle formed by the two lines is half of the (positive) difference of the measures of the intercepted arcs.
In this case, secant segments PB and PD are drawn to the circle from exterior point P, and the intercepted arcs are BD and CA. Therefore:
[tex]m\angle SPD = \dfrac{\overset{\frown}{BD}-\overset{\frown}{CA}}{2}[/tex]
Given that arc BD = 40° and arc CA = 15°, then:
[tex]m\angle SPD = \dfrac{40^{\circ}-15^{\circ}}{2}\\\\\\m\angle SPD = \dfrac{25^{\circ}}{2}\\\\\\m\angle SPD =12.5^{\circ}[/tex]
So, the measure of angle SPD is 12.5°.
[tex]\dotfill[/tex]
Question 8
To find the measure of angle RQT, we can use the Intersecting Tangents Theorem.
According to the theorem, if two tangents are drawn to the circle from one exterior point, the measure of the angle formed by the two lines is half of the (positive) difference of the measures of the intercepted arcs.
In this case, the two tangents intersect the circle at points R and T, and intersect each other at exterior point Q. The intercepted arcs are major arc RAT and minor arc RT. Therefore:
[tex]m\angle RQT= \dfrac{\overset{\frown}{RAT}-\overset{\frown}{RT}}{2}[/tex]
The sum of the measures of the arcs on a circle is always 360°. Therefore, the measure of major arc RAT is 360° less the measure of minor arc RT:
[tex]\overset{\frown}{RAT}=360^{\circ}-\overset{\frown}{RT}\\\\\overset{\frown}{RAT}=360^{\circ}-90^{\circ}\\\\\overset{\frown}{RAT}=270^{\circ}[/tex]
So, as arc RAT = 270° and arc RT = 90°, then:
[tex]m\angle RQT= \dfrac{270^{\circ}-90^{\circ}}{2}\\\\\\m\angle RQT= \dfrac{180^{\circ}}{2}\\\\\\m\angle RQT= 90^{\circ}[/tex]
So, the measure of angle RQT is 90°.
[tex]\dotfill[/tex]
Question 9
To find the measure of angle SRT, we can use the Tangent and Intersected Chord Theorem.
According to this theorem, if a tangent and a chord intersect at a point on a circle, the measure of each angle formed is one-half the measure of its intercepted arc.
In this case, chord TR and tangent RS intersect at point R on the circle. Therefore, the measure of angle SRT is half the measure of major arc RAT.
[tex]m\angle SRT = \dfrac{\overset{\frown}{RAT}}{2}[/tex]
Given that arc RAT = 270°, then:
[tex]m\angle SRT = \dfrac{270^{\circ}}{2}\\\\\\m\angle SRT =135^{\circ}[/tex]
So, the measure of angle SRT is 135°.
[tex]\dotfill[/tex]
Question 10
IMPORTANT: Please note that the provided diagram is not accurate. If we draw the diagram to scale using the given arc measures, the tangent at point T will not intersect the secants PB and PD at point P (see attachment). However, we will answer this question as if the tangent at point T does intersect secant PD at point P.
To find the measure of angle DPQ, we can use the Intersecting Secant-Tangent Theorem.
According to the theorem, if secant and a tangent are drawn to the circle from one exterior point, the measure of the angle formed by the two lines is half of the (positive) difference of the measures of the intercepted arcs.
In this case, secant PD and tangent PT are drawn to the circle from exterior point P, and the intercepted arcs are DRT and TC. Therefore:
[tex]m\angle DPQ= \dfrac{\overset{\frown}{DRT}-\overset{\frown}{TC}}{2}[/tex]
Given that arc DRT = 120° and arc TC = 75°, then:
[tex]m\angle DPQ= \dfrac{120^{\circ}-75^{\circ}}{2}\\\\\\m\angle DPQ= \dfrac{45^{\circ}}{2}\\\\\\m\angle DPQ= 22.5^{\circ}[/tex]
So, the measure of angle DPQ is 22.5°.
[tex]\dotfill[/tex]
Question 11
To find the measure of angle RTP, we can use the Tangent and Intersected Chord Theorem.
According to this theorem, if a tangent and a chord intersect at a point on a circle, the measure of each angle formed is one-half the measure of its intercepted arc.
In this case, chord TR and tangent PT intersect at point T on the circle. Therefore, the measure of angle RTP is half the measure of major arc RAT.
[tex]m\angle RTP= \dfrac{\overset{\frown}{RAT}}{2}[/tex]
Given that arc RAT = 270°, then:
[tex]m\angle RTP= \dfrac{270^{\circ}}{2}\\\\\\m\angle RTP=135^{\circ}[/tex]
So, the measure of angle RTP is 135°.
