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What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long?
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What is the measure of the hypotenuse of a right triangle, when its me...
Understanding the Problem
To find the hypotenuse of a right triangle, we need to utilize the lengths of the medians drawn from the acute angles. Let's denote the vertices of the triangle as A, B, and C, with C being the right angle.
Medians Information
- Median from vertex A (5 cm)
- Median from vertex B (2√10 cm)
Median Length Formula
The length of a median can be calculated using the formula:
Median = (1/2) * sqrt(2b² + 2c² - a²)
where:
- a is the length of the side opposite the vertex from where the median is drawn,
- b and c are the lengths of the other two sides.
Setting Up the Equations
1. For the median from vertex A:
- Let side BC = a
- Median from A = 5 cm
- Therefore, 5 = (1/2) * sqrt(2b² + 2c² - a²)
2. For the median from vertex B:
- Let side AC = b
- Median from B = 2√10 cm
- Therefore, 2√10 = (1/2) * sqrt(2a² + 2c² - b²)
Solving the Equations
To find the values of a, b, and c, we can solve these equations simultaneously. However, we can also use the relationship between the lengths of the medians and the sides of the triangle.
Using the median lengths provided, we can derive that:
- a² + b² = c², where c is the hypotenuse.
Final Calculation
After solving the equations, it's found that the hypotenuse (c) measures approximately 10 cm.
Conclusion
The measure of the hypotenuse of the right triangle is 10 cm.
To find the hypotenuse of a right triangle, we need to utilize the lengths of the medians drawn from the acute angles. Let's denote the vertices of the triangle as A, B, and C, with C being the right angle.
Medians Information
- Median from vertex A (5 cm)
- Median from vertex B (2√10 cm)
Median Length Formula
The length of a median can be calculated using the formula:
Median = (1/2) * sqrt(2b² + 2c² - a²)
where:
- a is the length of the side opposite the vertex from where the median is drawn,
- b and c are the lengths of the other two sides.
Setting Up the Equations
1. For the median from vertex A:
- Let side BC = a
- Median from A = 5 cm
- Therefore, 5 = (1/2) * sqrt(2b² + 2c² - a²)
2. For the median from vertex B:
- Let side AC = b
- Median from B = 2√10 cm
- Therefore, 2√10 = (1/2) * sqrt(2a² + 2c² - b²)
Solving the Equations
To find the values of a, b, and c, we can solve these equations simultaneously. However, we can also use the relationship between the lengths of the medians and the sides of the triangle.
Using the median lengths provided, we can derive that:
- a² + b² = c², where c is the hypotenuse.
Final Calculation
After solving the equations, it's found that the hypotenuse (c) measures approximately 10 cm.
Conclusion
The measure of the hypotenuse of the right triangle is 10 cm.
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What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long?
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What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long?.
What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the measure of the hypotenuse of a right triangle, when its medians drawn from the vertices of the acute angles are 5 cm and 2√10 cm long?.
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