Question

Reshma wishes to fit three rods together in the shape of a right triangle. The hypotenuse is to be 2 cm longer then the base and 4 cm longer than the altitude. What should be the lengths of the rods?

Answer 1

Answer :-

The lengths of the rods should be 6 cm, 8 cm and 10 cm.

Explanation :-

Let the length of the altitude of the right triangle be 'x' cm

Hypotenuse will be 4 cm longer than the altitude

So Length of the hypotenuse = (x + 4) cm

Hypotenuse is 2 cm longer than the base

i.e the base will be 2 cm shorter than the hypotenuse

So length of the base of the right triangle = {(x + 4) - 2} = (x + 4 - 2) = (x + 2) cm

By Pythagoras theorem

(Base)² + (Altitude)² = (Hypotenuse)²

⇒ (x + 2)² + x² = (x + 4)²

⇒ x² + 2² + 2(x)(2) + x² = x² + 4² + 2(x)(4)

⇒ 2x² + 4 + 4x = x² + 16 + 8x

⇒ 2x² + 4 + 4x - x² - 16 - 8x = 0

⇒ x² - 4x - 12 = 0

⇒ x² - 6x + 2x - 12 = 0

⇒ x(x - 6) + 2(x - 6) = 0

⇒ (x + 5)(x - 6) = 0

⇒ x + 5 = 0 or x - 6 = 0

⇒ x = - 5 or x = 6

We neglect x = - 5 as lengths cannot be negative

So x = 6 cm

i.e Length of the altitute of the right triangle = 6 cm

Length of the hypotenuse of the right triangle = (x + 4) = (6 + 4) = 10 cm

Length of the base of the right triangle = (x + 2) = (6 + 2) = 8 cm

So the lengths of the rods should be 6 cm, 8 cm and 10 cm.

Answer 2

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ATQ,

Let the altitude be 'x' cm

Hypotenuse is 'x+4' cm 

As the base is 2cm less than the hypotenuse, the base is 'x+2' cm

Now, as it is a right triangle, 

(x)2 + (x+2)2 = (x+4)2

=> x2 + (x2 +4x+4) = (x2 +8x+16)

=> 2x2 +4x +4 -x2 -8x-16=0

=> x2 -4x-12=0

=> x2 +2x-6x-12=0

=> x(x+2) -6 (x+2)=0

=> (x-6)(x+2)=0

Therefore,

x-6=0;  x+2=0

As side can’t be negative,

X=6 cm.

The sizes = x cm= 6 cm

    x+2 cm= 6+2= 8 cm

    x+4 cm=6+4= 10 cm