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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 Triangle ABC
Triangle ABC is a right-angled triangle with angle B measuring 90 degrees. The sides are given as follows:
- AB = 32
- BC = 24
Using the Pythagorean theorem, we can find the length of AC:
Calculating Side AC
The formula for the Pythagorean theorem is:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values:
\[ AC^2 = 32^2 + 24^2 \]
\[ AC^2 = 1024 + 576 \]
\[ AC^2 = 1600 \]
\[ AC = \sqrt{1600} = 40 \]
Now, we have the lengths of all sides:
- AC = 40
Finding the Circumcenter D
In a right triangle, the circumcenter (D) is located at the midpoint of the hypotenuse (AC). Therefore, we can find D by calculating the midpoint of AC:
- Coordinates of A = (0, 0)
- Coordinates of C = (32, 24)
The midpoint D is given by:
\[ D = \left(\frac{0 + 32}{2}, \frac{0 + 24}{2}\right) = (16, 12) \]
Finding Point E
A perpendicular line to AC through D intersects AB at point E. The slope of AC can be calculated as follows:
- Slope of AC = \(\frac{24-0}{32-0} = \frac{3}{4}\)
The slope of the perpendicular line (DE) is the negative reciprocal:
- Slope of DE = \(-\frac{4}{3}\)
Using point-slope form for line DE:
\[ y - 12 = -\frac{4}{3}(x - 16) \]
Setting \(y = 0\) to find E:
\[ 0 - 12 = -\frac{4}{3}(x - 16) \]
Solving for x gives \(x = 24\).
Final Calculation of BE
Since B is at (0, 0), the length BE is:
\[ BE = \sqrt{(24 - 0)^2 + (0 - 0)^2} = 24 \]
Thus, the value of BE is:
BE = 24
Triangle ABC is a right-angled triangle with angle B measuring 90 degrees. The sides are given as follows:
- AB = 32
- BC = 24
Using the Pythagorean theorem, we can find the length of AC:
Calculating Side AC
The formula for the Pythagorean theorem is:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values:
\[ AC^2 = 32^2 + 24^2 \]
\[ AC^2 = 1024 + 576 \]
\[ AC^2 = 1600 \]
\[ AC = \sqrt{1600} = 40 \]
Now, we have the lengths of all sides:
- AC = 40
Finding the Circumcenter D
In a right triangle, the circumcenter (D) is located at the midpoint of the hypotenuse (AC). Therefore, we can find D by calculating the midpoint of AC:
- Coordinates of A = (0, 0)
- Coordinates of C = (32, 24)
The midpoint D is given by:
\[ D = \left(\frac{0 + 32}{2}, \frac{0 + 24}{2}\right) = (16, 12) \]
Finding Point E
A perpendicular line to AC through D intersects AB at point E. The slope of AC can be calculated as follows:
- Slope of AC = \(\frac{24-0}{32-0} = \frac{3}{4}\)
The slope of the perpendicular line (DE) is the negative reciprocal:
- Slope of DE = \(-\frac{4}{3}\)
Using point-slope form for line DE:
\[ y - 12 = -\frac{4}{3}(x - 16) \]
Setting \(y = 0\) to find E:
\[ 0 - 12 = -\frac{4}{3}(x - 16) \]
Solving for x gives \(x = 24\).
Final Calculation of BE
Since B is at (0, 0), the length BE is:
\[ BE = \sqrt{(24 - 0)^2 + (0 - 0)^2} = 24 \]
Thus, the value of BE is:
BE = 24
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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.?
Question Description
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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