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Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.
Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is
Correct answer is '2'. Can you explain this answer?
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Let ABCD be a quadrilateral with area 18, with side AB parallel to th...
Given:
- ABCD is a quadrilateral with area 18.
- AB is parallel to CD and AB = 2CD.
- AD is perpendicular to AB and CD.
To find:
- The radius of the circle inscribed in quadrilateral ABCD.
Solution:
To find the radius of the inscribed circle, we need to determine the inradius of the quadrilateral ABCD.
Step 1: Divide the quadrilateral ABCD into two triangles.
- Let E be the point where the circle touches AB.
- Let F be the point where the circle touches CD.
Step 2: Calculate the area of each triangle.
- The area of triangle ADE is given by 1/2 * AD * DE.
- The area of triangle BCF is given by 1/2 * BC * CF.
Step 3: Calculate the areas of the two triangles.
- Since the area of the quadrilateral ABCD is 18, the sum of the areas of triangles ADE and BCF is also 18.
Step 4: Use the given information to express the areas of the two triangles in terms of the radius of the inscribed circle.
- Since the circle is tangent to all sides of the quadrilateral, the line segments AE, DE, BF, and CF are all radii of the circle.
- From the given information, we know that AB = 2CD.
- Therefore, DE = CF = r (radius of the circle), and AE = BF = 2r.
Step 5: Substitute the expressions for the areas of the two triangles into the equation from Step 3.
- The area of triangle ADE is given by 1/2 * AD * DE = 1/2 * AD * r.
- The area of triangle BCF is given by 1/2 * BC * CF = 1/2 * BC * r.
- The sum of the areas of triangles ADE and BCF is 18, so we have: 1/2 * AD * r + 1/2 * BC * r = 18.
Step 6: Use the given information to express AD and BC in terms of CD.
- Since AB = 2CD, we have AD = AB + BC = 2CD + BC.
Step 7: Substitute the expressions for AD and BC into the equation from Step 5.
- Substituting AD = 2CD + BC, the equation becomes: 1/2 * (2CD + BC) * r + 1/2 * BC * r = 18.
- Simplifying the equation: CD * r + BC * r + 1/2 * BC * r = 18.
- Factoring out r: r * (CD + BC + 1/2 * BC) = 18.
Step 8: Use the given information to express BC in terms of CD.
- Since AB = 2CD, we have BC = AB - CD = 2CD - CD = CD.
Step 9: Substitute the expression for BC into the equation from Step 7.
- Substituting BC = CD, the equation becomes: r * (CD +
- ABCD is a quadrilateral with area 18.
- AB is parallel to CD and AB = 2CD.
- AD is perpendicular to AB and CD.
To find:
- The radius of the circle inscribed in quadrilateral ABCD.
Solution:
To find the radius of the inscribed circle, we need to determine the inradius of the quadrilateral ABCD.
Step 1: Divide the quadrilateral ABCD into two triangles.
- Let E be the point where the circle touches AB.
- Let F be the point where the circle touches CD.
Step 2: Calculate the area of each triangle.
- The area of triangle ADE is given by 1/2 * AD * DE.
- The area of triangle BCF is given by 1/2 * BC * CF.
Step 3: Calculate the areas of the two triangles.
- Since the area of the quadrilateral ABCD is 18, the sum of the areas of triangles ADE and BCF is also 18.
Step 4: Use the given information to express the areas of the two triangles in terms of the radius of the inscribed circle.
- Since the circle is tangent to all sides of the quadrilateral, the line segments AE, DE, BF, and CF are all radii of the circle.
- From the given information, we know that AB = 2CD.
- Therefore, DE = CF = r (radius of the circle), and AE = BF = 2r.
Step 5: Substitute the expressions for the areas of the two triangles into the equation from Step 3.
- The area of triangle ADE is given by 1/2 * AD * DE = 1/2 * AD * r.
- The area of triangle BCF is given by 1/2 * BC * CF = 1/2 * BC * r.
- The sum of the areas of triangles ADE and BCF is 18, so we have: 1/2 * AD * r + 1/2 * BC * r = 18.
Step 6: Use the given information to express AD and BC in terms of CD.
- Since AB = 2CD, we have AD = AB + BC = 2CD + BC.
Step 7: Substitute the expressions for AD and BC into the equation from Step 5.
- Substituting AD = 2CD + BC, the equation becomes: 1/2 * (2CD + BC) * r + 1/2 * BC * r = 18.
- Simplifying the equation: CD * r + BC * r + 1/2 * BC * r = 18.
- Factoring out r: r * (CD + BC + 1/2 * BC) = 18.
Step 8: Use the given information to express BC in terms of CD.
- Since AB = 2CD, we have BC = AB - CD = 2CD - CD = CD.
Step 9: Substitute the expression for BC into the equation from Step 7.
- Substituting BC = CD, the equation becomes: r * (CD +
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Community Answer
Let ABCD be a quadrilateral with area 18, with side AB parallel to th...
9a2 + b2 - 6ab = a2 + b2
8a2 = 6ab
4a = 3b
Also, ab + 1/2ab = 3/2ab = 18
⇒ ab = 12
a = 3, b = 4
Radius = 2
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Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer?
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Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer?.
Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer?.
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Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer?, a detailed solution for Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer? has been provided alongside types of Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD.Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius isCorrect answer is '2'. Can you explain this answer? theory, EduRev gives you an
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