Question

The legs of a right triangle have the proportion $8:15$, and the hypotenuse is $6.8cm$ long. Find the area of the triangle $?$

Answer 1

Find the area of the right triangle:

Step- 1: Identify the smaller side of the right triangle:

The given ratio of the length of legs is $8˸15$ and the length of hypotenuse is $6.8cm$

Let the length of the smaller leg be $8x$ and the length of larger leg be $15x$.

Step- 2 : Use Pythagorean theorem to evaluate:

${\left[\text{Hypotenuse}\right]}^2={\left[\text{Smaller leg}\right]}^2+{\left[\text{Larger leg}\right]}^2$

$\Rightarrow$ ${(6.8)}^2={\left(8x\right)}^2+{\left(15x\right)}^2$

$\Rightarrow$ $46.24=64x^2+225x^2$

$\Rightarrow$ $289x^2=46.24$

Step- 3 : Take the square root of both the side to find the value of $x$:

$17x=6.8$

$\therefore$ $x=0.4$

So, the length of smaller leg $=8x$

$$\begin{gathered} =8\times 0.4 \\ =3.2\text{cm} \end{gathered}$$

Length of larger leg $=15x$

$$\begin{gathered} =15\times 0.4 \\ =6\text{cm} \end{gathered}$$

[Area of right triangle is half the product of lengths of its legs.]

Area $=\frac{1}{2}\times 6\times 3.2$

$=9.6\text{square centimeters}$

Hence, the area of right triangle is $\mathbf{9}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{\text{square centimeters}}$.