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PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?
Correct answer is '144'. Can you explain this answer?
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PQRS is a quadrilateral which is inscribed in a circle. Measures of th...
Let the angles of the quadrilateral be 2a, 3a, 4a and b.
Since PQRS is a cyclic quadrilateral, the sum of its opposite angles is 180°.
Three cases are possible.
Case I: 2a + 3a = 4a + b= 180°
Since PQRS is a cyclic quadrilateral, the sum of its opposite angles is 180°.
Three cases are possible.
Case I: 2a + 3a = 4a + b= 180°
5a = 180° which implies that a = 36° and b = 180 - 144 = 36° So, the angles are 72°, 108°, 144° and 36°.
Case II:
Case II:
2a + 4a = 3a + b = 180°
6a = 180° which implies that 3a = 90°
Since 3a is one of the angles of the quadrilateral, this implies that in this case one of the diagonals of the quadrilateral is a diameter of the circumcircle. Hence, this case is discarded.
Case III:
Case III:
2a + b = 3a + 4a = 180°
7a = 180° which implies that a « 25.71°
So, the angles are approximately of measure 51.42°, 77.13°, 102.84° and 128.58°.
Thus, the largest possible angle is 144°.
Answer: 144
Thus, the largest possible angle is 144°.
Answer: 144
Most Upvoted Answer
PQRS is a quadrilateral which is inscribed in a circle. Measures of th...
The given problem states that PQRS is a quadrilateral inscribed in a circle. Let's solve this problem step by step:
Step 1: Understand the problem.
We are given a quadrilateral PQRS inscribed in a circle. We need to find the measure of the largest possible angle.
Step 2: Draw a diagram.
Draw a circle and inscribe the quadrilateral PQRS in it. Label the angles as given in the problem.
Step 3: Understand the properties of an inscribed quadrilateral.
In an inscribed quadrilateral, opposite angles are supplementary, i.e., they add up to 180 degrees.
Step 4: Use the given information.
The measures of three of the angles are in the ratio 2:3:4. Let's assume the measures of the angles to be 2x, 3x, and 4x.
Step 5: Apply the properties of an inscribed quadrilateral.
Using the property mentioned in Step 3, we can write the equation:
2x + 4x = 180 (opposite angles are supplementary)
6x = 180
x = 30
Step 6: Find the measures of the angles.
Using the value of x, we can find the measures of the angles:
Angle P = 2x = 2(30) = 60 degrees
Angle Q = 3x = 3(30) = 90 degrees
Angle R = 4x = 4(30) = 120 degrees
Angle S = 180 - (Angle R + Angle P + Angle Q) = 180 - (120 + 60 + 90) = 180 - 270 = -90 degrees
Step 7: Analyze the answer.
The measure of Angle S is -90 degrees, which is not possible for an angle in a quadrilateral. We made an error in our calculations.
Step 8: Redo the calculations.
Since Angle S cannot be negative, we need to find an alternative value for the measure of Angle S. Let's try the next possible value for x.
Step 9: Redo the calculations with a different value for x.
Let's assume x = 40.
Angle P = 2x = 2(40) = 80 degrees
Angle Q = 3x = 3(40) = 120 degrees
Angle R = 4x = 4(40) = 160 degrees
Angle S = 180 - (Angle R + Angle P + Angle Q) = 180 - (160 + 80 + 120) = 180 - 360 = -180 degrees
Step 10: Analyze the answer.
Again, the measure of Angle S is negative, which is not possible for an angle in a quadrilateral. We need to try a larger value for x.
Step 11: Redo the calculations with a larger value for x.
Let's assume x = 50.
Angle P = 2x = 2(50) = 100 degrees
Angle Q = 3x = 3(50) = 150 degrees
Angle R = 4x = 4(50) =
Step 1: Understand the problem.
We are given a quadrilateral PQRS inscribed in a circle. We need to find the measure of the largest possible angle.
Step 2: Draw a diagram.
Draw a circle and inscribe the quadrilateral PQRS in it. Label the angles as given in the problem.
Step 3: Understand the properties of an inscribed quadrilateral.
In an inscribed quadrilateral, opposite angles are supplementary, i.e., they add up to 180 degrees.
Step 4: Use the given information.
The measures of three of the angles are in the ratio 2:3:4. Let's assume the measures of the angles to be 2x, 3x, and 4x.
Step 5: Apply the properties of an inscribed quadrilateral.
Using the property mentioned in Step 3, we can write the equation:
2x + 4x = 180 (opposite angles are supplementary)
6x = 180
x = 30
Step 6: Find the measures of the angles.
Using the value of x, we can find the measures of the angles:
Angle P = 2x = 2(30) = 60 degrees
Angle Q = 3x = 3(30) = 90 degrees
Angle R = 4x = 4(30) = 120 degrees
Angle S = 180 - (Angle R + Angle P + Angle Q) = 180 - (120 + 60 + 90) = 180 - 270 = -90 degrees
Step 7: Analyze the answer.
The measure of Angle S is -90 degrees, which is not possible for an angle in a quadrilateral. We made an error in our calculations.
Step 8: Redo the calculations.
Since Angle S cannot be negative, we need to find an alternative value for the measure of Angle S. Let's try the next possible value for x.
Step 9: Redo the calculations with a different value for x.
Let's assume x = 40.
Angle P = 2x = 2(40) = 80 degrees
Angle Q = 3x = 3(40) = 120 degrees
Angle R = 4x = 4(40) = 160 degrees
Angle S = 180 - (Angle R + Angle P + Angle Q) = 180 - (160 + 80 + 120) = 180 - 360 = -180 degrees
Step 10: Analyze the answer.
Again, the measure of Angle S is negative, which is not possible for an angle in a quadrilateral. We need to try a larger value for x.
Step 11: Redo the calculations with a larger value for x.
Let's assume x = 50.
Angle P = 2x = 2(50) = 100 degrees
Angle Q = 3x = 3(50) = 150 degrees
Angle R = 4x = 4(50) =
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PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer?
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PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer?.
PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer?.
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PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer?, a detailed solution for PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer? has been provided alongside types of PQRS is a quadrilateral which is inscribed in a circle. Measures of three of its angles are in the ratio of 2 : 3 : 4. If any of the diagonals of the quadrilateral is not the diameter of the circumcircle, what is the measure of the largest possible angle?Correct answer is '144'. Can you explain this answer? theory, EduRev gives you an
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