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ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC is
- a)1:2
- b)2:3
- c)3:4
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
ABC is a right angled triangle, B being the right angle. Midpoints of ...
Formulas:
Area of a triangle 
Area of a trapezium =
Triangle ABC and A’B’C are similar by SAS
By property of similar triangles,
∠B = ∠A’B’C and 2A’B’ = AB
∴ A’B’ is parallel to AB.
Thus quadrilateral AA’ B’B is a trapezium.
In quadrilateral AA’ B’B,
Height = BB’
⇒ Height = BC/2
Sum of parallel sides = (A’B’ + AB) = 3AB/2
Area of quadrilateral AA’ B’B
⇒ Area of quadrilateral AA’ B’B
Area of triangle ABC
Ratio area of quadrilateral AA’ B’B to area of triangle ABC
⇒ Ratio of area of quadrilateral AA’ B’B to area of triangle ABC = 3 : 4
Area of a trapezium =
Triangle ABC and A’B’C are similar by SAS
By property of similar triangles,
∠B = ∠A’B’C and 2A’B’ = AB
∴ A’B’ is parallel to AB.
Thus quadrilateral AA’ B’B is a trapezium.
In quadrilateral AA’ B’B,
Height = BB’
⇒ Height = BC/2
Sum of parallel sides = (A’B’ + AB) = 3AB/2
Area of quadrilateral AA’ B’B
⇒ Area of quadrilateral AA’ B’B
Area of triangle ABC
Ratio area of quadrilateral AA’ B’B to area of triangle ABC
⇒ Ratio of area of quadrilateral AA’ B’B to area of triangle ABC = 3 : 4
Most Upvoted Answer
ABC is a right angled triangle, B being the right angle. Midpoints of ...
1 and M2. Let P be the midpoint of AB. We need to prove that
(i) AM1 = BM2 = PM1 = PM2
(ii) Quadrilateral AM1PB is a square.
Proof:
(i) As P is the midpoint of AB, we have AP = PB. Also, as M1 and M2 are midpoints of BC and AC respectively, we have BM1 = CM1 and AM2 = CM2.
Since ABC is a right angled triangle, we have BM2 = BC/2 and AM1 = AC/2. Therefore, to prove AM1 = BM2, we need to prove that AC = BC. But this is true because ABC is a right angled triangle with B as the right angle.
To prove PM1 = PM2, we use the fact that P is the midpoint of AB. Therefore, PM1 is the median of triangle PBC and PM2 is the median of triangle PAC. But as ABC is a right angled triangle, we have BP = PC and AP = PC. Therefore, PM1 = PM2.
(ii) To prove that AM1PB is a square, we need to prove that all its sides are equal and the diagonals are equal and perpendicular to each other.
As we have already proved that AM1 = BM2 and PM1 = PM2, it follows that AM1PB is a rhombus with equal sides AM1, BM1, PM1, and PM2.
To prove that the diagonals are equal and perpendicular, we use the fact that P is the midpoint of AB and M1 and M2 are midpoints of BC and AC respectively. Therefore, PM1 is perpendicular to BC and PM2 is perpendicular to AC. Also, PB is perpendicular to BC and PA is perpendicular to AC. Therefore, PB || AC and PA || BC. Therefore, PM1 and PM2 are the perpendicular bisectors of AB and PB and PA are the perpendicular bisectors of BC.
Therefore, the diagonals of AM1PB are PB and PM1 and they are equal and perpendicular to each other. Hence, AM1PB is a square.
(i) AM1 = BM2 = PM1 = PM2
(ii) Quadrilateral AM1PB is a square.
Proof:
(i) As P is the midpoint of AB, we have AP = PB. Also, as M1 and M2 are midpoints of BC and AC respectively, we have BM1 = CM1 and AM2 = CM2.
Since ABC is a right angled triangle, we have BM2 = BC/2 and AM1 = AC/2. Therefore, to prove AM1 = BM2, we need to prove that AC = BC. But this is true because ABC is a right angled triangle with B as the right angle.
To prove PM1 = PM2, we use the fact that P is the midpoint of AB. Therefore, PM1 is the median of triangle PBC and PM2 is the median of triangle PAC. But as ABC is a right angled triangle, we have BP = PC and AP = PC. Therefore, PM1 = PM2.
(ii) To prove that AM1PB is a square, we need to prove that all its sides are equal and the diagonals are equal and perpendicular to each other.
As we have already proved that AM1 = BM2 and PM1 = PM2, it follows that AM1PB is a rhombus with equal sides AM1, BM1, PM1, and PM2.
To prove that the diagonals are equal and perpendicular, we use the fact that P is the midpoint of AB and M1 and M2 are midpoints of BC and AC respectively. Therefore, PM1 is perpendicular to BC and PM2 is perpendicular to AC. Also, PB is perpendicular to BC and PA is perpendicular to AC. Therefore, PB || AC and PA || BC. Therefore, PM1 and PM2 are the perpendicular bisectors of AB and PB and PA are the perpendicular bisectors of BC.
Therefore, the diagonals of AM1PB are PB and PM1 and they are equal and perpendicular to each other. Hence, AM1PB is a square.
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ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer?.
ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer? for SSC 2024 is part of SSC preparation. The Question and answers have been prepared according to the SSC exam syllabus. Information about ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for SSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABC is a right angled triangle, B being the right angle. Midpoints of BC and AC are respectively B’ and A’. The ratio of the area of the quadrilateral AA’ B’B to the area of the triangle ABC isa)1:2b)2:3c)3:4d)None of theseCorrect answer is option 'C'. Can you explain this answer?.
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