Question

The difference between the sides at right angled triangle is 14 cm. The area of the triangle is 120cm sq. Calculate the lenght of hypotenuse of the triangle

Answer 1

$\textbf{Answer}$ :

26 cm

$\textbf{Step by Step Explanation}$ :

We know that, the two sides of a right angled triangle are its height and base respectively.

Here, we are given that the difference between the sides is 14 cm. Let one of the sides be x. Then, the second side will be x + 14.

Also, it is given that the area of the triangle is 120 sq. cms. The area of a right angled triangle is given by

ar(right angled triangle) = 1/2 × measure of base of the triangle × measure of height of the triangle

So,

$\implies \frac{1}{2} \times x \times (x + 14) = 120\\ \\ \implies \: \frac{1}{2} \times {x}^{2} + 14x = 120 \\ \\ \implies \: \frac{ {x}^{2} + 14x}{2} = 120 \\ \\ \implies \: {x}^{2} + 14x = 240 \\ \\ \implies \: {x}^{2} + 14x - 240 = 0 \\ \\ \implies \: {x}^{2} + 24x - 10x - 240 = 0 \\ \\ \implies \: x(x + 24) - 10(x + 24) = 0 \\ \\ \implies \: (x + 24)(x - 10) = 0 \\ \\ \implies \: x = 10 \: or \: - 24$

But, it is known that the magnitude of length can never be negative. Thus, length of one side is 10 cm and the other is 24 cm ( which was x + 14).

Now, we need to find the value of the hypotenuse of the triangle. For this, we will apply Pythagoras Theorem.

$\textbf{Pythagoras Theorem}$ :

The Pythagoras Theorem states that the square of the hypotenuse of a right angled triangle is the sum of the square of the other two sides, i.e.

${h}^{2} = {a}^{2} + {b}^{2}$

Putting in the values of the two sides we found earlier, we get

${h}^{2} = {10}^{2} + {24}^{2} \\ \\ {h}^{2} = 100 + 576 \\ \\ {h}^{2} = 676 \\ \\ h = \sqrt{2 \times 2 \times 13 \times 13} \\ \\ h = 2 \times 13 \\ \\ h = 26 \: cm$

Hence, the length of the hypotenuse is 26 cm.

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Answer 2

The other answerer Mylo had solved it quite well. I will give you solutions in other method. Check it out!

So, Given that the difference between the sides of the right angled triangle is 14cm.

Let us consider the sides are x, y

So, x - y = 14

Note that These two are sides of the right angled triangle other than its hypotenuse.

Now, Also given that Area of triangle = 120sq.cm.

We know that,

Area of right triangle = 1/2 * Foot * Perpendicular.

120 = 1/2 * x * y

240 = xy.

Using, ( a + b)² = (a - b) ² + 4ab.

Here,

$(x+y) ^{2} = (x-y) ^{2} + 4xy \\ \\ = (14) ^{2} +4(240) \\ \\ = 196 + 960 \\ \\ = 1156 \\ \\ {(x + y)}^{2} = 1156 \\ \\ (x + y) = \sqrt{1156} = 34$

Now,
x + y = 34
x - y = 14

Adding both of them gives, 2x = 48, x = 24

Substituting value of x gives y = 10.

Now,

Applying Pythagoras theorem.

$Hypotenuse = \sqrt{(24 ^{2} + {10}^{2} ) } \\\\= \sqrt{676} = 26$


Therefore, Length of the hypotenuse is 26 cm.