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In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle?
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Problem:
In a figure, a circle with center O is inscribed in a quadrilateral ABCD such that, it touches the sides AB, BC, AD, CD at points P, Q, R, S respectively. If AB = 29 cm, AD = 23 cm, angle B = 90°, DS = 5 cm, find the radius of the circle.
Solution:
Step 1: Draw the figure
Draw a quadrilateral ABCD with right angle at B. Mark the points of contact of the circle with the sides as P, Q, R, and S.
Step 2: Identify the given information
From the problem statement, we are given:
- AB = 29 cm
- AD = 23 cm
- Angle B = 90°
- DS = 5 cm
Step 3: Identify the relevant properties
To find the radius of the inscribed circle, we can make use of the following properties:
- The radius of the inscribed circle is perpendicular to the side it touches.
- The radius of the inscribed circle is equidistant from the points of contact.
Step 4: Apply the properties to solve the problem
Let the radius of the circle be r. Since the radius is perpendicular to the side it touches, we can draw perpendiculars from O to AB and AD, intersecting at X and Y respectively.
Step 5: Use Pythagoras Theorem
Using Pythagoras Theorem in triangle ABX, we have:
- AX^2 + BX^2 = AB^2
- (r + r)^2 + (AB - 2r)^2 = AB^2
- 4r^2 + (AB - 2r)^2 = AB^2
- 4r^2 + (29 - 2r)^2 = 29^2
Step 6: Solve the equation
Expanding the equation and simplifying, we get:
- 4r^2 + 841 - 116r + 4r^2 = 841
- 8r^2 - 116r = 0
- 4r(2r - 29) = 0
Step 7: Determine the value of r
Since the radius cannot be zero, we have:
- 2r - 29 = 0
- 2r = 29
- r = 29/2
- r = 14.5 cm
Step 8: Answer
The radius of the inscribed circle is 14.5 cm.
In a figure, a circle with center O is inscribed in a quadrilateral ABCD such that, it touches the sides AB, BC, AD, CD at points P, Q, R, S respectively. If AB = 29 cm, AD = 23 cm, angle B = 90°, DS = 5 cm, find the radius of the circle.
Solution:
Step 1: Draw the figure
Draw a quadrilateral ABCD with right angle at B. Mark the points of contact of the circle with the sides as P, Q, R, and S.
Step 2: Identify the given information
From the problem statement, we are given:
- AB = 29 cm
- AD = 23 cm
- Angle B = 90°
- DS = 5 cm
Step 3: Identify the relevant properties
To find the radius of the inscribed circle, we can make use of the following properties:
- The radius of the inscribed circle is perpendicular to the side it touches.
- The radius of the inscribed circle is equidistant from the points of contact.
Step 4: Apply the properties to solve the problem
Let the radius of the circle be r. Since the radius is perpendicular to the side it touches, we can draw perpendiculars from O to AB and AD, intersecting at X and Y respectively.
Step 5: Use Pythagoras Theorem
Using Pythagoras Theorem in triangle ABX, we have:
- AX^2 + BX^2 = AB^2
- (r + r)^2 + (AB - 2r)^2 = AB^2
- 4r^2 + (AB - 2r)^2 = AB^2
- 4r^2 + (29 - 2r)^2 = 29^2
Step 6: Solve the equation
Expanding the equation and simplifying, we get:
- 4r^2 + 841 - 116r + 4r^2 = 841
- 8r^2 - 116r = 0
- 4r(2r - 29) = 0
Step 7: Determine the value of r
Since the radius cannot be zero, we have:
- 2r - 29 = 0
- 2r = 29
- r = 29/2
- r = 14.5 cm
Step 8: Answer
The radius of the inscribed circle is 14.5 cm.
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In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle?
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In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle? 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 a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle?.
In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle? 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 a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle?.
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In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle?, a detailed solution for In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle? has been provided alongside types of In a figure, a circle with center O is inscribed in a quadrilateral abcd such that, it touches the sidesAB, BC,AD,CD at points P,Q,R,S respectively . If ab=29 cm, ad=23cm angle b = 90 , ds=5cm , find the radius of the circle? theory, EduRev gives you an
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