Question

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

Answer 1

Let the altitude be x cm

Then

The hypotenuse is (x + 4) cm [hypotenuse is 4 cm longer than the altitude]

The base is ((x + 4) - 2) cm = (x + 2) cm [hypotenuse is 2 cm less than the base]

According to the right angle triangle law

Square of the hypotenuse = Sum of the squares of the other two sides.

=> (x + 4)^2 = x^2 + (x + 2)^2

=> x^2 + 16 + 8x = x^2 + x^2 + 4 +4x

=> x^2 + 16 + 8x = 2x^2 + 4 +4x

=> 2x^2 + 4 +4x - (x^2 + 16 + 8x) = 0

=> 2x^2 + 4 +4x - x^2 -16 -8x = 0

=> x^2 -4x -12 = 0

=> x^2 -6x +2x -12 = 0

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

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

=> x - 6 = 0, x + 2 = 0

=> x = 6, -2

As a side cannot be negative; x = 6 cm =>Altitude is 6 cm.

The base = (x + 2) cm = (6 + 2) cm => 8 cm

The hypotenuse = (x + 4) cm => (6 + 4) cm => 10 cm

Therefore, the lengths of the rods are 6 cm, 8 cm and 10 cm.

Answer 2

Let the Altitude of a right triangle be x Given:Hypotenuse of a right triangle is 4 cm longer than the Altitude, then
hypotenuse is (x + 4) cm

Given: Hypotenuse is 2 cm longer than the base,then
Hypotenuse= base + 2
x+4 = base + 2
x + 4 -2 = base
x +2 = base
Base = (x + 2) cm

According to Pythagoras theorem: H² = P² +B²
(x + 4)² = x² + (x + 2)²
x²+ 16 + 8x = x² + x² + 4 +4x
[(a+b)²= a² +2ab +b²]
x² + 16 + 8x = 2x² + 4 +4x
2x² + 4 +4x - x² - 16 - 8x = 0
x² - 4x -12 = 0
x² -6x +2x -12 = 0 [by Middle term splitting]
x(x - 6) +2(x - 6) = 0
(x - 6) (x + 2) = 0
x - 6 = 0 or x + 2 = 0
x = 6 or x= -2
side cannot be negative ,so x = 6 cm Altitude (x) = 6 cm.
Base = (x + 2) cm = (6 + 2) cm = 8 cm
hypotenuse = (x + 4) cm = (6 + 4) cm = 10 cm
Hence,the lengths of the rods are 6 cm, 8 cm and 10 cm.

HOPE THIS WILL HELP YOU...