Question

length of hypotenuse of a right angle triangle exceeds twice the length of the altitude by 3 cm find the length of each of the triangle

Answer 1

let the length of altitude be x

therefore hypotenuse =2x+3 cm

Answer 2

Let base = x cm

Hypotenuse = (x+2)cm

Given,

Hypotenuse = 2 × altitude + 1 cm

$\longrightarrow$ $\sf{x+2=2×altitude+1cm}$

$\longrightarrow$ $\sf{altitude=\frac{1}{2}(x+1)cm}$

By Pythagoras theorem,

$\large\sf\purple{{hypotenuse}^{2} = {base}^{2} + {altitude}^{2}}$

$\longrightarrow$$\sf\orange{ ({x + 2)}^{2} = {x}^{2} + \frac{1}{4} {(x + 1)}^{2}}$

$\longrightarrow$$\sf\orange{4( {x}^{2} + 4x + 4) = {4x}^{2} + ( {x}^{2} + 2x + 1)}$

$\longrightarrow$$\sf\orange{{x}^{2} - 14x - 15 = 0}$

$\longrightarrow$$\sf\orange{(x - 15)(x + 1) = 0}$

$\longrightarrow$$\sf\orange{x = 15 \: or \: x = - 1}$

As length cannot be negative, x ≠ -1, so x = 15

Hence,

base = 15cm

Hypotenuse = 15+2 = 17cm

Altitude = $\sf{\frac{1}{2}(15+1)=8cm}$