Question

The legs of a right triangle are in the ratio 3:4 and its area is $1014 \text{cm}^{2}$. Find the length of its legs.

Answer 1

Hint:
We are given the ratio of sides of a right triangle as $3:4$ and its area as $1014 \text{cm}^{2}$. We will suppose the base and height be $3x$ and $4x$ respectively and then by the formula of the triangle as $\dfrac{1}{2} \times {\text{base}} \times {\text{height}}$, we will put the value of the sides and equate it to the given area of the triangle. Upon simplifying the obtained equation for the value of x, we will get the value of x and hence the length of the sides (legs) of the triangle.

Complete step by step solution:
We have the ratio of the legs (sides) of the triangle given in the question as $3: 4$
The area of the triangle is given = $1014 \text{cm}^{2}$
Now, let the length of the sides be $3x$ and $4x$.
We know that the formula of the area of the triangle is given by the formula: $\dfrac{1}{2} \times {\text{base}} \times {\text{height}}$
Substituting the values of the sides, we get
$ \Rightarrow $ Area of the triangle = $\dfrac{1}{2} \times {\text{base}} \times {\text{height}}$
$ \Rightarrow $Area of the triangle = $\dfrac{1}{2} \times 3x \times 4x$
$ \Rightarrow $Area of the triangle = $\dfrac{1}{2} \times 12{x^2} = 6{x^2}$
Now, the area of the triangle given in the question is $1014 \text{cm}^{2}$.
Substituting the value of the area of the triangle in the obtained equation, we get
$
   \Rightarrow 1014 = 6{x^2} \\
   \Rightarrow 169 = {x^2} \\
   \Rightarrow x = \sqrt {169} = 13 \\
 $
Therefore, the length of $x$ is $13$ cm.
Hence, putting this value of x to determine the length of the sides of the given right triangle, we get
$ \Rightarrow 3x = 3 (13) = 39$ cm, and $4x = 4 (13) = 52$ cm.

Hence, the length of the legs of the triangle are 39cm and 52cm respectively.

Note:
Just to understand the concept of ratio better take it as in each 7 cm length, 3 cm will be the base and 4 cm will be the height. but it’s in the ratio and we know this ratio is similar to $6:8$ or $9:12$. Also they are nothing but multiples. That’s why we have multiplied $3$ and $4$ with $x$. Then we found the correct multiple with the help of the given condition.