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A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)?
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A circle is touching the side BC of triangle ABC at P and touching AB ...
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A circle is touching the side BC of triangle ABC at P and touching AB ...
Problem
A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)?
Solution
Step 1: Draw a diagram
The first step to solve the problem is to draw a diagram. We have to draw a triangle ABC and a circle that touches the side BC at point P and touches the sides AB and AC produced at points Q and R respectively.
Step 2: Use the tangent property of circles
We know that the tangent to a circle is perpendicular to the radius at the point of contact. Therefore, we can draw perpendiculars from points Q and R to the line BC, as shown in the diagram. Let the length of the perpendicular from Q be x and the length of the perpendicular from R be y.
Step 3: Use the properties of tangents
We know that the tangents from a point to a circle are equal in length. Therefore, we can write the following equations:
QA = QP + PA
RA = RP + PA
Since the circle is tangent to BC at P, we know that PA = PC. Therefore:
QA = QP + PC
RA = RP + PC
Step 4: Use the properties of triangles
We know that the sum of the lengths of the sides of a triangle is equal to its perimeter. Therefore, we can write:
AB + BC + AC = perimeter of triangle ABC
We also know that BC = BP + PC. Therefore, we can write:
AB + BP + PC + AC = perimeter of triangle ABC
Step 5: Use the properties of the circle
We know that the circle touches AB and AC produced at points Q and R respectively. Therefore, AQ = AB + x and AR = AC + y.
Step 6: Substitute values
We can substitute the values of QA, RA, AQ, and AR into the equation we derived in step 4. This gives us:
AB + BP + QP + PC + AC + RP + PC = perimeter of triangle ABC
Simplifying this equation, we get:
AB + AC + 2PC + 2x + 2y = perimeter of triangle ABC
Substituting the values of AQ and AR from step 5, we get:
AB + AC + 2PC + AQ - AB + AR - AC = perimeter of triangle ABC
Simplifying this equation, we get:
2PC + 2x + 2y = perimeter of triangle ABC
Step 7: Simplify the equation
We know that PC = (perimeter of triangle ABC)/2 - BC/2. Substituting this value into the equation we derived in step 6, we get:
(perimeter of triangle ABC)/2 - BC/2 + x + y = perimeter of triangle ABC/2
Simplifying this equation, we get:
x + y = (perimeter of triangle ABC)/2 - BC/2
Step 8: Substitute values
We know that x and y are the lengths of the perpendiculars from Q and R to BC
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A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)?
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A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)? 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 A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)?.
A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)? 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 A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle is touching the side BC of triangle ABC at P and touching AB and AC produced at Q and R respectively. Prove that AQ = 1/2 (perimeter of triangle ABC)?.
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