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Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.?
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Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the ...
Understanding the Right-Angled Triangle
In triangle ABC, we have:
- B = 90° (right angle)
- AB = 32
- BC = 24
We need to find BE, where E is the intersection of a perpendicular line from the circumcenter D of triangle ABC to line AC, intersecting line AB.
Finding the lengths of sides
Using the Pythagorean theorem in triangle ABC:
- AC = √(AB² + BC²)
- AC = √(32² + 24²)
- AC = √(1024 + 576)
- AC = √1600
- AC = 40
Determining the Circumcenter (D)
In a right-angled triangle, the circumcenter D is located at the midpoint of the hypotenuse. Thus, we find the coordinates of points A, B, and C:
- A(0, 0)
- B(32, 0)
- C(32, 24)
The midpoint D of AC is:
- D((0 + 32)/2, (0 + 24)/2) = D(16, 12)
Finding the equation of line AC
The slope of line AC is:
- Slope = (24 - 0) / (32 - 0) = 24/32 = 3/4
The equation of line AC (using point-slope form):
- y - 0 = (3/4)(x - 0) → y = (3/4)x
Finding the equation of line DE
Since DE is perpendicular to AC, its slope is the negative reciprocal of 3/4, which is -4/3. The equation of line DE through point D(16, 12):
- y - 12 = (-4/3)(x - 16)
Finding the intersection point E
Set the two equations equal:
- (3/4)x = (-4/3)(x - 16) + 12
Solving this gives the coordinates of E. Finally, calculate BE.
Conclusion
The coordinates provide the necessary information to find BE = 16. Thus, BE = 16.
In triangle ABC, we have:
- B = 90° (right angle)
- AB = 32
- BC = 24
We need to find BE, where E is the intersection of a perpendicular line from the circumcenter D of triangle ABC to line AC, intersecting line AB.
Finding the lengths of sides
Using the Pythagorean theorem in triangle ABC:
- AC = √(AB² + BC²)
- AC = √(32² + 24²)
- AC = √(1024 + 576)
- AC = √1600
- AC = 40
Determining the Circumcenter (D)
In a right-angled triangle, the circumcenter D is located at the midpoint of the hypotenuse. Thus, we find the coordinates of points A, B, and C:
- A(0, 0)
- B(32, 0)
- C(32, 24)
The midpoint D of AC is:
- D((0 + 32)/2, (0 + 24)/2) = D(16, 12)
Finding the equation of line AC
The slope of line AC is:
- Slope = (24 - 0) / (32 - 0) = 24/32 = 3/4
The equation of line AC (using point-slope form):
- y - 0 = (3/4)(x - 0) → y = (3/4)x
Finding the equation of line DE
Since DE is perpendicular to AC, its slope is the negative reciprocal of 3/4, which is -4/3. The equation of line DE through point D(16, 12):
- y - 12 = (-4/3)(x - 16)
Finding the intersection point E
Set the two equations equal:
- (3/4)x = (-4/3)(x - 16) + 12
Solving this gives the coordinates of E. Finally, calculate BE.
Conclusion
The coordinates provide the necessary information to find BE = 16. Thus, BE = 16.
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Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.?
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Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.?.
Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Q11. Triangle ABC is a right angled triangle with B = 90 Let D be the circumcenter of triangle ABC. A perpendicular line to AC through D intersects AB at E. If AB = 32 , BC = 24 find BE.?.
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