Question

The altitude of a right triangle is 7cm less than its base.if the hypotenuse is 13cm ,find the other two sides​

Answer 1

Given :

• Hypotenuse of the Right-angled triangle = 13.

• Height of the Right-angled triangle is 7 cm less than the base of the triangle.

To find :

The side of the triangle.

Solution :

Let the base of the triangle be x cm , so according to the question , the other side will be (x - 7).

Hence, we get the values of height and base (in terms of x) as :-

• Base = x cm

• Height = (x - 7) cm

Now by using the Pythagoras theorem , we can find the value of x.

Pythagoras theorem :-

$\underline{\boxed{\bf{H^{2} = P^{2} - B^{2}}}}$

Where :-

• H = Hypotenuse

• P = Height

• B = Base

Now using the formula and substituting the values in it, we get :-

$:\implies \bf{H^{2} = P^{2} + B^{2}}$

$:\implies \bf{13^{2} = (x - 7)^{2} + x^{2}}$

Now by using the Equation ,

$\bf{(a - b)^{2} = a^{2} - 2ab + b^{2}}$ , we get :-

$:\implies \bf{13^{2} = x^{2} - 2 \times 7 \times x + 7^{2} + x^{2}}$

$:\implies \bf{13^{2} = x^{2} - 14x + 7^{2} + x^{2}}$

$:\implies \bf{13^{2} = 2x^{2} - 14x + 7^{2}}$

$:\implies \bf{169 = 2x^{2} - 14x + 49}$

$:\implies \bf{0 = 2x^{2} - 14x + 49 - 169}$

$:\implies \bf{0 = 2x^{2} - 14x - 120}$

$:\implies \bf{0 = 2x^{2} - (24 - 10)x - 120}$

$:\implies \bf{0 = 2x^{2} - 24x + 10x - 120}$

$:\implies \bf{0 = 2x(x - 12) + 10(x - 12)}$

$:\implies \bf{0 = (x - 12)(2x + 10})$

$\underline{\therefore \bf{(x - 12)(2x + 10) = 0}}$

Now to find the value of x.

• $:\implies \bf{(2x + 10) = 0}$

$:\implies \bf{2x = -10}$

$:\implies \bf{x = \dfrac{-10}{2}}$

$:\implies \bf{x = - 5}$

Hence, the value of x is - 5.

• $:\implies \bf{(x - 12) = 0}$

$:\implies \bf{x = 12}$

Hence, the value of x is 12.

But we know that the height can't be in negative , hence the value of x is 12 cm.

Thus, the base of the triangle is 12 cm.

Now , we know that the Height is 7 cm lesser than the base , i.e,

$:\implies \bf{h = (x - 7)}$

$:\implies \bf{h = (12 - 7)}\:\:[\because Base = 12\:cm]$

$:\implies \bf{h = 5}$

$\underline{\therefore \bf{Height\:(h) = 5\:cm}}$

Hence, the base of the triangle is 5 cm.

Answer 2

here is ur answer mate.........