Question

In a triangle ABC, the sides are in the ratio 3 : 4 : 5. If the area of the triangle is 96 sq units, what is its perimeter?

Hint:

Recognizing Pythagorean triplets like (3, 4, 5), (5, 12, 13), (8, 15, 17), etc., can save a lot of time by immediately identifying a triangle as right-angled, which simplifies area calculations.

Answer 1

Correct Answer: 48

Step 1: Understanding the Question:
The question provides the ratio of the sides of a triangle and its area. We need to find the perimeter. The ratio 3:4:5 is a special case known as a Pythagorean triplet, which means the triangle is a right-angled triangle. This information simplifies the area calculation.
Step 2: Key Formula or Approach:
1. Sides of the triangle: Let the sides be \(3x\), \(4x\), and \(5x\) for some positive constant \(x\).
2. Area of a right-angled triangle: Since the sides form a Pythagorean triplet (\((3x)^2 + (4x)^2 = 9x^2 + 16x^2 = 25x^2 = (5x)^2\)), the triangle is right-angled. The two shorter sides, \(3x\) and \(4x\), are the base and height. The area is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] 3. Perimeter of a triangle: The perimeter is the sum of the lengths of its sides.
\[ \text{Perimeter} = 3x + 4x + 5x = 12x \] Step 3: Detailed Explanation:
Let the lengths of the sides of the triangle be \(3x, 4x,\) and \(5x\).
The area of the triangle is given as 96 sq units. Using the area formula for a right-angled triangle:
\[ \text{Area} = \frac{1}{2} \times (3x) \times (4x) = 96 \] \[ \frac{1}{2} \times 12x^2 = 96 \] \[ 6x^2 = 96 \] Now, we solve for \(x\):
\[ x^2 = \frac{96}{6} \] \[ x^2 = 16 \] \[ x = \sqrt{16} = 4 \] (We take the positive value since \(x\) represents a scaling factor for length).
Now that we have the value of \(x\), we can find the perimeter of the triangle.
\[ \text{Perimeter} = 12x \] \[ \text{Perimeter} = 12 \times 4 = 48 \text{ units} \] The lengths of the sides are \(3(4)=12\), \(4(4)=16\), and \(5(4)=20\). The perimeter is \(12+16+20 = 48\).
Step 4: Final Answer:
The perimeter of the triangle is 48 units.