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A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²?
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A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then...
Solution:
Given:
- Triangle ABC is inscribed in a semicircle of radius 4.
- AB = 8
Approach:
- Let O be the center of the semicircle and D be the midpoint of AB.
- Since triangle ABC is inscribed in a semicircle, AC and BC are both radii of the semicircle, hence AC = BC = 4.
- We need to find the maximum value of (AC + BC)² = (4 + 4)² = 64.
Explanation:
- As AC = BC = 4, triangle ABC is an isosceles triangle with base AB = 8.
- The perpendicular from O to AB bisects AB at D, making AD = BD = 4.
- Therefore, triangle AOD is a right-angled triangle with AO = OD = 4 and AD = 4.
- By Pythagoras theorem, AD² + AO² = OD²
- 4² + 4² = OD²
- 16 + 16 = OD²
- OD² = 32
- OD = √32 = 4√2
- Therefore, the height of triangle ABC is 4√2.
- The area of triangle ABC is 1/2 * base * height = 1/2 * 8 * 4√2 = 16√2.
- Since the area of triangle ABC is constant, the maximum value of (AC + BC)² = 64.
Therefore, the maximum value of (AC + BC)² is 64.
Given:
- Triangle ABC is inscribed in a semicircle of radius 4.
- AB = 8
Approach:
- Let O be the center of the semicircle and D be the midpoint of AB.
- Since triangle ABC is inscribed in a semicircle, AC and BC are both radii of the semicircle, hence AC = BC = 4.
- We need to find the maximum value of (AC + BC)² = (4 + 4)² = 64.
Explanation:
- As AC = BC = 4, triangle ABC is an isosceles triangle with base AB = 8.
- The perpendicular from O to AB bisects AB at D, making AD = BD = 4.
- Therefore, triangle AOD is a right-angled triangle with AO = OD = 4 and AD = 4.
- By Pythagoras theorem, AD² + AO² = OD²
- 4² + 4² = OD²
- 16 + 16 = OD²
- OD² = 32
- OD = √32 = 4√2
- Therefore, the height of triangle ABC is 4√2.
- The area of triangle ABC is 1/2 * base * height = 1/2 * 8 * 4√2 = 16√2.
- Since the area of triangle ABC is constant, the maximum value of (AC + BC)² = 64.
Therefore, the maximum value of (AC + BC)² is 64.
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A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²?
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A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²?.
A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A triangle ABC is inscribed in a semicircle of radius 4. If AB=8, then what is the maximum value of (AC+BC)²?.
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