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Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?
(1) ∠QPD = 135°
(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2 – 10x + 1 = 0
(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2
(1) ∠QPD = 135°
(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2 – 10x + 1 = 0
(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2
Select the correct answer using the code given below.
- a)1 and 2 only
- b)2 and 3 only
- c)1 and 3 only
- d)1, 2 and 3
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider a circle with centre at O and radius 7 cm. Let QR be the chor...
In triangle OPC
⇒ ∠OPC = 90° [∵ Perpendicular from centre bisects the chord]
⇒ OP2 = OQ2 – PQ2 [∵ Pythagoras theorem]
⇒ OP = √(72 – 12) = √48 cm
⇒ ∠OPC = 90° [∵ Perpendicular from centre bisects the chord]
⇒ OP2 = OQ2 – PQ2 [∵ Pythagoras theorem]
⇒ OP = √(72 – 12) = √48 cm
In triangle OPM
∠OMP = 90° [∵ Perpendicular from centre bisects the chord]
⇒ OM2 = OP2 – PM2 [∵ Pythagoras theorem]
⇒ OM = √(48 – 24) = √24 cm
∠OMP = 90° [∵ Perpendicular from centre bisects the chord]
⇒ OM2 = OP2 – PM2 [∵ Pythagoras theorem]
⇒ OM = √(48 – 24) = √24 cm
In triangle OMC
∠OMC = 90° [∵ Perpendicular from centre bisects the chord]
⇒ MC2 = OC2 – OM2 [∵ Pythagoras theorem]
⇒ MC = √(72 – 24) = √25 = 5cm
∠OMC = 90° [∵ Perpendicular from centre bisects the chord]
⇒ MC2 = OC2 – OM2 [∵ Pythagoras theorem]
⇒ MC = √(72 – 24) = √25 = 5cm
Statement I:
In triangle OPM
OM = PM = √24
⇒ It is an isosceles right triangle
⇒ ∠OPM = 45°
∠RPM = ∠RPO + ∠OPM
⇒ ∠RPM = 90 + 45 = 135°
∠QPD = ∠RPM = 135° [∵ Vertically opposite angles]
⇒ Statement I is true.
In triangle OPM
OM = PM = √24
⇒ It is an isosceles right triangle
⇒ ∠OPM = 45°
∠RPM = ∠RPO + ∠OPM
⇒ ∠RPM = 90 + 45 = 135°
∠QPD = ∠RPM = 135° [∵ Vertically opposite angles]
⇒ Statement I is true.
Statement II:
CP, m = MC + MP = 5 + √24 cm
PD, n = MD – PM = 5 – √24 cm [∵ MC = MD]
⇒ Roots of the equations 5 ± √24
⇒ Sum of roots = 10
⇒ Product of roots = (5 + √24) (5 – √24) = 25 – 24 = 1
⇒ The quadratic equation = x2 – 10x + 1 = 0
⇒ Statement II is true
CP, m = MC + MP = 5 + √24 cm
PD, n = MD – PM = 5 – √24 cm [∵ MC = MD]
⇒ Roots of the equations 5 ± √24
⇒ Sum of roots = 10
⇒ Product of roots = (5 + √24) (5 – √24) = 25 – 24 = 1
⇒ The quadratic equation = x2 – 10x + 1 = 0
⇒ Statement II is true
Statement III:
Area of OPR/Area of OMP = (1/2 × PR × OP)/(1/2 × OM × PM)
⇒ (1/2 × 1 × √48)/(1/2 × √24 × √24)
⇒ √[48/ (24 × 24)]
⇒ 1/√12 = 1 ∶ 2√3
⇒ Statement III is wrong
∴ Only Statement 1 and 2 is correct.
Area of OPR/Area of OMP = (1/2 × PR × OP)/(1/2 × OM × PM)
⇒ (1/2 × 1 × √48)/(1/2 × √24 × √24)
⇒ √[48/ (24 × 24)]
⇒ 1/√12 = 1 ∶ 2√3
⇒ Statement III is wrong
∴ Only Statement 1 and 2 is correct.
Most Upvoted Answer
Consider a circle with centre at O and radius 7 cm. Let QR be the chor...
Given Information:
- Circle with center O and radius 7 cm
- Chord QR with length 2 cm and midpoint P
- Chord CD passing through P, with midpoint M and MP = √24 cm
Verification of Statements:
Statement 1: ∠QPD = 135°
- Since P is the midpoint of QR, we can conclude that ∠QPO is a right angle
- ∠QPO = 90° (angle at the center is twice the angle at the circumference)
- This means ∠QPD = 135°, as it is the supplementary angle to ∠QPO
- Therefore, statement 1 is correct.
Statement 2: CP = m cm and PD = n cm, m and n are the roots of x^2 - 10x + 1 = 0
- Let CP be x and PD be y
- Since P is the midpoint of QR, CP = PD = 1 cm (half of QR length)
- Therefore, x = y = 1 cm
- The quadratic equation x^2 - 10x + 1 = 0 does not have roots as 1, 1
- Hence, statement 2 is incorrect.
Statement 3: Ratio of triangle OPR to triangle OMP is 1:2√2
- Area of triangle OPR = (1/2) * 7 * 7 = 24.5 sq cm
- Area of triangle OMP = (1/2) * 7 * √24 = 14√6 sq cm
- The ratio of triangle OPR to triangle OMP is approximately 24.5 : 14√6, not 1 : 2√2
- Therefore, statement 3 is incorrect.
Conclusion:
- Only statement 1 is correct.
- Option A is the correct answer.
- Circle with center O and radius 7 cm
- Chord QR with length 2 cm and midpoint P
- Chord CD passing through P, with midpoint M and MP = √24 cm
Verification of Statements:
Statement 1: ∠QPD = 135°
- Since P is the midpoint of QR, we can conclude that ∠QPO is a right angle
- ∠QPO = 90° (angle at the center is twice the angle at the circumference)
- This means ∠QPD = 135°, as it is the supplementary angle to ∠QPO
- Therefore, statement 1 is correct.
Statement 2: CP = m cm and PD = n cm, m and n are the roots of x^2 - 10x + 1 = 0
- Let CP be x and PD be y
- Since P is the midpoint of QR, CP = PD = 1 cm (half of QR length)
- Therefore, x = y = 1 cm
- The quadratic equation x^2 - 10x + 1 = 0 does not have roots as 1, 1
- Hence, statement 2 is incorrect.
Statement 3: Ratio of triangle OPR to triangle OMP is 1:2√2
- Area of triangle OPR = (1/2) * 7 * 7 = 24.5 sq cm
- Area of triangle OMP = (1/2) * 7 * √24 = 14√6 sq cm
- The ratio of triangle OPR to triangle OMP is approximately 24.5 : 14√6, not 1 : 2√2
- Therefore, statement 3 is incorrect.
Conclusion:
- Only statement 1 is correct.
- Option A is the correct answer.
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Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer?
Question Description
Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer?.
Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer?.
Solutions for Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Defence.
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Here you can find the meaning of Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer?, a detailed solution for Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider a circle with centre at O and radius 7 cm. Let QR be the chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct?(1) ∠QPD = 135°(2) If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x2– 10x + 1 = 0(3) The ratio of triangle OPR to the area of triangle OMP is 1 ∶ 2√2Select the correct answer using the code given below.a)1 and 2 onlyb)2 and 3 onlyc)1 and 3 onlyd)1, 2 and 3Correct answer is option 'A'. Can you explain this answer? tests, examples and also practice Defence tests.
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