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In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp?
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In figure PQ is tangent to the circle at R and AB is diameter of the c...
In the given figure, we have a circle with diameter AB and a tangent line PQ, which touches the circle at point R. We are given that angle PQR is equal to 30 degrees, and we need to find the measure of angle ARP.
Properties of a Tangent Line:
1. A tangent line to a circle is perpendicular to the radius drawn to the point of tangency.
2. The angle between a tangent line and a radius is 90 degrees.
Identify the Key Points:
Let's identify the key points in the figure for better understanding:
- Point A: One end of the diameter AB.
- Point B: The other end of the diameter AB.
- Point P: A point on the tangent line PQ.
- Point Q: The other end of the tangent line PQ.
- Point R: The point of tangency between the circle and the tangent line.
- Point O: The center of the circle.
Analysis:
Since PQ is tangent to the circle at R, angle PQR is 90 degrees (property 2). We are given that angle PQR is 30 degrees, so angle QRP is 60 degrees (as the sum of angles in a triangle is 180 degrees).
To find angle ARP, we need to consider triangle APR:
- Angle ARP is the angle opposite to side PR.
- Angle ARP + angle PRA + angle APR = 180 degrees (sum of angles in a triangle).
Solution:
From the analysis, we know that angle QRP is 60 degrees. Since the tangent line is perpendicular to the radius, angle PRA is also 90 degrees (property 1). Therefore, angle APR can be found as follows:
Angle APR + angle PRA + angle QRP = 180 degrees
Angle APR + 90 degrees + 60 degrees = 180 degrees
Angle APR + 150 degrees = 180 degrees
Angle APR = 180 degrees - 150 degrees
Angle APR = 30 degrees
Hence, the measure of angle ARP is 30 degrees.
Properties of a Tangent Line:
1. A tangent line to a circle is perpendicular to the radius drawn to the point of tangency.
2. The angle between a tangent line and a radius is 90 degrees.
Identify the Key Points:
Let's identify the key points in the figure for better understanding:
- Point A: One end of the diameter AB.
- Point B: The other end of the diameter AB.
- Point P: A point on the tangent line PQ.
- Point Q: The other end of the tangent line PQ.
- Point R: The point of tangency between the circle and the tangent line.
- Point O: The center of the circle.
Analysis:
Since PQ is tangent to the circle at R, angle PQR is 90 degrees (property 2). We are given that angle PQR is 30 degrees, so angle QRP is 60 degrees (as the sum of angles in a triangle is 180 degrees).
To find angle ARP, we need to consider triangle APR:
- Angle ARP is the angle opposite to side PR.
- Angle ARP + angle PRA + angle APR = 180 degrees (sum of angles in a triangle).
Solution:
From the analysis, we know that angle QRP is 60 degrees. Since the tangent line is perpendicular to the radius, angle PRA is also 90 degrees (property 1). Therefore, angle APR can be found as follows:
Angle APR + angle PRA + angle QRP = 180 degrees
Angle APR + 90 degrees + 60 degrees = 180 degrees
Angle APR + 150 degrees = 180 degrees
Angle APR = 180 degrees - 150 degrees
Angle APR = 30 degrees
Hence, the measure of angle ARP is 30 degrees.
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In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp?
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In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp?.
In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In figure PQ is tangent to the circle at R and AB is diameter of the circle If angle=30 find angle arp?.
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