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In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is?
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In a right angle triangle with legs 4 and 8 the area of the largest sq...
**Introduction:**
In this problem, we are given a right-angled triangle with legs measuring 4 and 8. We need to find the area of the largest square that can be inscribed in this triangle. To solve this problem, we will use geometry and algebraic techniques.
**Approach:**
To find the largest square inscribed in a right-angled triangle, we need to determine the length of the square's side. We will follow these steps to find the side length and area of the square:
1. **Determine the Hypotenuse:** Using the Pythagorean theorem, we can find the length of the hypotenuse of the right-angled triangle. Let's denote it by 'c'.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, we have:
c² = 4² + 8²
c² = 16 + 64
c² = 80
c = √80
Therefore, the length of the hypotenuse is √80.
2. **Find the Altitude:** The altitude of the right-angled triangle is the perpendicular distance from the right angle to the hypotenuse. Let's denote it by 'h'.
Using the formula for the altitude of a right-angled triangle, we have:
h = (leg₁ * leg₂) / hypotenuse
h = (4 * 8) / √80
h = 32 / √80
h = (32/√80) * (√80/√80)
h = 32√80 / 80
h = √80 / 2
Therefore, the altitude of the triangle is √80 / 2.
3. **Determine the Side Length of the Square:** Since the square is inscribed in the right-angled triangle, its side will be equal to the altitude of the triangle.
Therefore, the side length of the square is √80 / 2.
4. **Calculate the Area of the Square:** The area of a square is given by the formula: area = side².
Substituting the value of the side length, we have:
area = (√80 / 2)²
area = 80 / 4
area = 20
Therefore, the area of the largest square inscribed in the triangle is 20 square units.
**Conclusion:**
In conclusion, the largest square that can be inscribed in a right-angled triangle with legs measuring 4 and 8 has an area of 20 square units. This problem demonstrates the application of geometry and algebraic techniques to solve for the side length and area of the square.
In this problem, we are given a right-angled triangle with legs measuring 4 and 8. We need to find the area of the largest square that can be inscribed in this triangle. To solve this problem, we will use geometry and algebraic techniques.
**Approach:**
To find the largest square inscribed in a right-angled triangle, we need to determine the length of the square's side. We will follow these steps to find the side length and area of the square:
1. **Determine the Hypotenuse:** Using the Pythagorean theorem, we can find the length of the hypotenuse of the right-angled triangle. Let's denote it by 'c'.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus, we have:
c² = 4² + 8²
c² = 16 + 64
c² = 80
c = √80
Therefore, the length of the hypotenuse is √80.
2. **Find the Altitude:** The altitude of the right-angled triangle is the perpendicular distance from the right angle to the hypotenuse. Let's denote it by 'h'.
Using the formula for the altitude of a right-angled triangle, we have:
h = (leg₁ * leg₂) / hypotenuse
h = (4 * 8) / √80
h = 32 / √80
h = (32/√80) * (√80/√80)
h = 32√80 / 80
h = √80 / 2
Therefore, the altitude of the triangle is √80 / 2.
3. **Determine the Side Length of the Square:** Since the square is inscribed in the right-angled triangle, its side will be equal to the altitude of the triangle.
Therefore, the side length of the square is √80 / 2.
4. **Calculate the Area of the Square:** The area of a square is given by the formula: area = side².
Substituting the value of the side length, we have:
area = (√80 / 2)²
area = 80 / 4
area = 20
Therefore, the area of the largest square inscribed in the triangle is 20 square units.
**Conclusion:**
In conclusion, the largest square that can be inscribed in a right-angled triangle with legs measuring 4 and 8 has an area of 20 square units. This problem demonstrates the application of geometry and algebraic techniques to solve for the side length and area of the square.
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In a right angle triangle with legs 4 and 8 the area of the largest sq...
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In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is?.
In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a right angle triangle with legs 4 and 8 the area of the largest square that can be inscribed in the triangle is?.
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