Finding the Sides of a 3:4:5 Triangle Given Its Area

Knowing the area of a triangle can help us determine the actual lengths of its sides, especially when the sides are in a known ratio. In this article, we'll explore how to find the sides of a 3:4:5 triangle when its area is given as 24 cm2.

Understanding the 3:4:5 Triangle

A 3:4:5 triangle is a special right triangle, where the lengths of the sides are in the ratio 3:4:5. It is considered a Pythagorean triple because it satisfies the equation (3^2 4^2 5^2), making it a right triangle.

Using the Pythagorean Theorem

Let's denote the sides of the triangle as 3x, 4x, and 5x, where x is a common multiplier.

Application of the Pythagorean Theorem

Since the ratio 3:4:5 implies a right triangle, we can use the Pythagorean Theorem to confirm this:

(3x^2 4x^2 5x^2)

This equation is true for any positive value of x, confirming that the triangle is right-angled.

Calculating the Area

The area of a right triangle can be found by using the formula:

(A frac{1}{2} times text{base} times text{height})

For our 3:4:5 triangle, we can choose 3x and 4x as the base and height, respectively:

(A frac{1}{2} times 3x times 4x 6x^2)

Solving for x

We are given that the area is 24 cm2. Therefore, we can set up the equation:

(6x^2 24)

Solving for x:

(x^2 frac{24}{6} 4)

(x sqrt{4} 2)

The Lengths of the Sides

With x 2, we can now find the lengths of the sides:

Side 1: (3x 3 times 2 6) cm Side 2: (4x 4 times 2 8) cm Side 3: (5x 5 times 2 10) cm

Therefore, the sides of the triangle are 6 cm, 8 cm, and 10 cm.

Using the Area Formula Directly

We can also verify the area using the direct formula for the area of a right triangle:

(A frac{1}{2} times 3x times 4x 6x^2)

Substituting x 2:

(A 6 times 2^2 6 times 4 24) cm2

This confirms that the area is indeed 24 cm2.

A Lookup Table for 3:4:5 Triangle

For future reference, here is a table summarizing the sides and area of a 3:4:5 triangle when the multiplier is 2:

Sides Area (cm2) 3x 6 cm 4x 8 cm 5x 10 cm A 24 cm2

Conclusion

By using the properties of a 3:4:5 triangle and the Pythagorean Theorem, we can find the actual lengths of the sides when the area is given. In this case, the sides of the triangle are 6 cm, 8 cm, and 10 cm, and the area is 24 cm2.