Question

A right-angled triangle has a width of x cm. The length of the hypotenuse is 10 cm. The perimeter of the triangle is 24cm. Find the area of the triangle

Answer 1

The area of the right-angled triangle is
`\( 24 \, \text{cm}^2 \)`.

The length of the hypotenuse is given as 10 cm. The perimeter of the triangle is the sum of the three sides:

`\[ \text{Perimeter} = \text{Width} + \text{Height} + \text{Hypotenuse} \]`

Given that the perimeter is 24 cm, we can set up the equation:

`\[ x + \text{Height} + 10 = 24 \]`

Now, solve for the height
`(\(\text{Height}\))`:

`\[ \text{Height} = 24 - x - 10 \]`

`\[ \text{Height} = 14 - x \]`

Now, we can use the Pythagorean theorem to relate the width, height, and hypotenuse:

`\[ x^2 + (\text{Height})^2 = \text{Hypotenuse}^2 \]`

Substitute the expression for height:

`\[ x^2 + (14 - x)^2 = 10^2 \]`

Now, solve for x:

`\[ x^2 + (196 - 28x + x^2) = 100 \]`

`\[ 2x^2 - 28x + 96 = 0 \]`

Divide the entire equation by 2 to simplify:

`\[ x^2 - 14x + 48 = 0 \]`

Now, factor the quadratic equation:

`\[ (x - 6)(x - 8) = 0 \]`

So,
`\( x = 6 \)` or
`\( x = 8 \)`. Since the width cannot be negative, we take
`\( x = 8 \)` cm.

Now that we have the width
`(\( x = 8 \))` and the height
`(\( \text{Height} = 14 - x = 14 - 8 = 6 \))`, we can find the area
`(\( A \))` of the right-angled triangle using the formula:

`\[ A = (1)/(2) * \text{Width} * \text{Height} \]`

Substitute the values:

`\[ A = (1)/(2) * 8 * 6 \]`

`\[ A = 24 \]`

Therefore, the area of the right-angled triangle is
`\( 24 \, \text{cm}^2 \)`.