Question

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

Answer 1

Given:-

• The altitude of a right triangle is 7cm less then it's base the hypotenus is 13 cm.

To find:-

• find the other two side.

Solutions:-

• Let the base of the right triangle be x cm.

• it's altitude = (x - 7) cm

From Pythagoras theorem;

• Base² + Altitude ² = Hypothesis ²

Therefore,

=> x² + (x - 7)² = 13²

=> x² + x² + 49 - 14x = 169

=> 2x² - 14x - 169 - 49 = 0

=> 2x² - 14x - 120 = 0

=> 2(x² - 7x - 60) = 0

=> x² - 7x - 60 = 0

=> x² - 12x + 5x - 60 = 0

=> x(x - 12) + 5(x - 12) = 0

=> (x - 12) (x + 5) = 0

=> x - 12 = 0 or x + 5 = 0

=> x = 12 0r x = -5

Since,

• sides are positive x can only be 12.

Therefore,

The base of the triangle is 12cm and altitude of the triangle be (12 - 7)cm = 5cm.

Hence, the others two sides is 12cm and 5cm.

Answer 2

Given,

• Altitude of right triangle is 7 cm less than its base.
• Hypotenuse is 13 cm.

To find,

• The other two sides.

Solution,

• Let x be the base of the triangle
• Then altitude will be (x-7)

We know that,

$\sf{Base^2+Altitude^2=Hypotenuse^2}$

So, by pythagoras theorem,

${x}^{2} + ( {x - 7)}^{2} = {13}^{2} \\ \\ ⟹2 {x}^{2} - 14x + 49 = 169 \\ \\ ⟹2 {x}^{2} - 14x + 49 - 169 = 0 \\ \\ ⟹2 {x}^{2} - 14x - 120 = 0 \\ \\ ⟹2( {x}^{2} - 7x - 60) = 0 \\ \\ ⟹ {x}^{2} - 7x - 60 = \frac{0}{2} \\ \\⟹ {x }^{2} - 7x - 60 = 0 \\ \\ ⟹ {x}^{2} - 12x + 5x - 60 = 0 \\ \\ ⟹x(x - 12) + 5(x - 12) = 0 \\ \\ ⟹(x - 12)(x + 5) = 0$

So, x = 12 or x = -5

Since,the side of a triangle cannot be negative,so the base of the triangle is 12 cm.

And the altitude will be (12-7) = 5 cm