Understanding the Right-Angled Triangle Problem

The problem provides information about a right-angled triangle: the ratio of its side lengths is 5 : 12 : 13, and its perimeter is 90 cm. We need to find the area of this triangle.

The ratio 5 : 12 : 13 is a classic Pythagorean triplet, which confirms that the triangle is indeed right-angled. In a right-angled triangle with sides in this ratio, the sides corresponding to 5 and 12 are the perpendicular legs (base and height), and the side corresponding to 13 is the hypotenuse (the longest side, opposite the right angle).

Calculating the Actual Side Lengths

Let the common ratio factor be $x$. So, the lengths of the sides of the triangle are $5x$, $12x$, and $13x$.

The perimeter of a triangle is the sum of its side lengths. We are given that the perimeter is 90 cm.

Perimeter $= 5x + 12x + 13x$

Perimeter $= (5 + 12 + 13)x$

Perimeter $= 30x$

We know the perimeter is 90 cm, so we can set up the equation:

$30x = 90$

To find the value of $x$, divide both sides by 30:

$x = \frac{90}{30}$

$x = 3$

Now that we have the value of $x$, we can calculate the actual lengths of the sides:

• Side 1: $5x = 5 \times 3 = 15$ cm
• Side 2: $12x = 12 \times 3 = 36$ cm
• Side 3: $13x = 13 \times 3 = 39$ cm

The side lengths are 15 cm, 36 cm, and 39 cm.

Finding the Area of the Right Triangle

For a right-angled triangle, the area can be calculated using the formula:

Area $= \frac{1}{2} \times \text{base} \times \text{height}$

In a right-angled triangle, the base and height are the two shorter sides (the legs that form the right angle). From the side lengths we calculated (15 cm, 36 cm, 39 cm), the two shorter sides are 15 cm and 36 cm. The longest side, 39 cm, is the hypotenuse.

Let's take the base as 15 cm and the height as 36 cm (or vice versa, the result will be the same).

Area $= \frac{1}{2} \times 15 \text{ cm} \times 36 \text{ cm}$

Area $= \frac{1}{2} \times (15 \times 36) \text{ cm}^2$

Area $= \frac{1}{2} \times 540 \text{ cm}^2$

Area $= 270 \text{ cm}^2$

So, the area of the right-angled triangle is 270 cm².

Revision Table: Key Concepts

Additional Information: Pythagorean Triplets and Area

The ratio 5 : 12 : 13 is a fundamental Pythagorean triplet. This means that if a right-angled triangle has sides proportional to these numbers, then $ (5k)^2 + (12k)^2 = (13k)^2 $ for any positive number $k$. In our case, $k$ is the factor $x$ we found, which is 3.

The sides are $5 \times 3 = 15$, $12 \times 3 = 36$, and $13 \times 3 = 39$. Let's check the Pythagorean theorem:

$15^2 + 36^2 = 225 + 1296 = 1521$

$39^2 = 1521$

Since $15^2 + 36^2 = 39^2$, the sides indeed form a right-angled triangle. The legs are 15 cm and 36 cm, and the hypotenuse is 39 cm.

The area formula Area $= \frac{1}{2} \times \text{base} \times \text{height}$ is applicable to all triangles, but for a right triangle, the legs serve directly as the base and height, making the calculation straightforward once the side lengths are known.