4.2.5: The altitude of a right angle triangle is 7 cm less than its base, If the hypotenuse is 13 cm, find the other two sides.

Question

(Q) The altitude of a right angle triangle is 7 cm less than its base, If the hypotenuse is 13 cm, find the other two sides.

Sapling Academy · Accepted Answer

Let's the height of the triangle be $x$

By given condition, it is 7 cm less than it base,

or we can say that, its base is 7 cm more than the height

hence height will be $x + 7$

Next, it is given that the hypotenuse is 13 cm

We know that by Pythagoras theorem, Height 2 + Base 2 = Hypotenuse 2

$\Rightarrow (x^2) + (x + 7)^2 = (13)^2$

$\Rightarrow (x^2) + (x^2 + 14x + 49) = 169$

$\Rightarrow x^2 + x^2 + 14x + 49 = 169$

$\Rightarrow 2 x^2 + 14x - 120 = 0 $

$\Rightarrow  x^2 + 7x - 60 = 0 $

$\Rightarrow  x^2 + 12 x - 5 x + 60 = 0 $

$\Rightarrow  x (x + 12) - 5 (x + 12) = 0 $

$\Rightarrow  (x + 12) (x - 5) = 0 $

$\Rightarrow  x = -12 ~ and ~ x = 5 $

Here, we reject $x = - 12$ because the value of the base can not be a negative number.

Therefore, $x = 5$ and $x + 7 = 5 + 7 = 12$

Hence, the base of the triangle is 5 cm and height of the triangle is 7 cm.

Check:

If sides are 5 cm and 12 cm, let's check the given condition:

(5)2 + (12)2 = (13)2  or 25 + 144 = 169

Since the equation is satisfied, hence our answer is correct.

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