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In a right angled triangle with integral sides, the ratio of the area to perimeters is 3:2. What is the perimeter of the smallest such triangle? A. 24 B. 36 C. 54 D. 56?
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In a right angled triangle with integral sides, the ratio of the area ...
Problem Analysis:
Let the sides of the right angled triangle be a, b and c, where c is the hypotenuse. The area of the triangle is given by 1/2 * a * b, and the perimeter is given by a + b + c. We are given that the ratio of the area to perimeter is 3:2. Therefore, we have the equation:

(1/2 * a * b) / (a + b + c) = 3/2

Simplifying this equation, we get:

ab = 3(a + b + c)

We need to find the smallest integral values of a, b and c that satisfy this equation.

Solution:

Step 1: Start by assuming a value for a. Let's assume a = 1.

Step 2: Substitute the value of a into the equation ab = 3(a + b + c).

1b = 3(1 + b + c)

Simplifying, we get:

b = 3 + 3b + 3c

2b = 3 + 3c

Step 3: Rearrange the equation to isolate b.

2b - 3b = 3c - 3

-b = 3c - 3

Step 4: Divide both sides by -1 to make b positive.

b = 3 - 3c

Step 5: Substitute the value of b into the equation ab = 3(a + b + c).

1(3 - 3c) = 3(1 + (3 - 3c) + c)

Simplifying, we get:

3 - 3c = 3(4 - 2c)

3 - 3c = 12 - 6c

Step 6: Rearrange the equation to isolate c.

3c - 6c = 12 - 3

-3c = 9

c = -3/3

Since the sides of the triangle must be positive integers, the value of c is not valid.

Step 7: Try a different value for a. Let's assume a = 2.

Step 8: Repeat steps 2-6 using the new value of a.

2b = 3 + 3c

b = (3 + 3c) / 2

2(3 + 3c) = 3(2 + (3 + 3c) + c)

6 + 6c = 3(5 + 4c)

6 + 6c = 15 + 12c

6c - 12c = 15 - 6

-6c = 9

c = -9/6

Again, the value of c is not valid.

Step 9: Repeat steps 7-8 with different values of a until a valid integral solution is found.

Step 10: The smallest valid integral solution is a = 3, b = 4, c = 5.

Step 11: Calculate the perimeter of the triangle.

Perimeter = a + b + c = 3 + 4 + 5
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In a right angled triangle with integral sides, the ratio of the area to perimeters is 3:2. What is the perimeter of the smallest such triangle? A. 24 B. 36 C. 54 D. 56?
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In a right angled triangle with integral sides, the ratio of the area to perimeters is 3:2. What is the perimeter of the smallest such triangle? A. 24 B. 36 C. 54 D. 56? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about In a right angled triangle with integral sides, the ratio of the area to perimeters is 3:2. What is the perimeter of the smallest such triangle? A. 24 B. 36 C. 54 D. 56? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a right angled triangle with integral sides, the ratio of the area to perimeters is 3:2. What is the perimeter of the smallest such triangle? A. 24 B. 36 C. 54 D. 56?.
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