QUESTION:

The hypotenuse of a right angled triangle is 9 cm more than the shortest SIDE. The THIRD side is 8 cm less than the hypotenuse, find the SIDES of the triangle.

ANSWER:

The sides are

• Hypotenuse = 29 cm
• Shortest side = 20 cm
• Third side = 21 cm.

GIVEN:

• The hypotenuse of a right angled triangle is 9 cm more than the shortest side.

• The third side is 8 cm less than the hypotenuse.

EXPLANATION:

Let the shortest side be x cm.

Hypotenuse = x + 9 cm.

Third side = Hypotenuse - 8

Third side = x + 9 - 8

Third side = x + 1 cm

According to Pythagoras theorem:

Square of hypotenuse is equal to the sum of the squares of the other two sides.

(x + 9)² = x² + (x + 1)²

(A + B)² = A² + 2AB + B²

x² + 2(9x) + 9² = x² + x² + 2(x) + 1²

x² + 18x + 81 = 2x² + 2x + 1

x² - 16x - 80 = 0

Splitting the middle term

x² - 16x - 4x + 4x - 80 = 0

x² - 20 x + 4x - 80 = 0

x(x - 20) + 4(x - 20)

(x + 4)(x - 20) = 0

x = 20 (Length cannot be negative so x ≠ - 4)

Length of shortest side = 20 cm

Hypotenuse = x + 9 = 20 + 9 = 29 cm

Third side = x + 1 = 20 + 1 = 21 cm.

VERIFICATION:

(x + 9)² = x² + (x + 1)²

(20 + 9)² = 20² + (20 + 1)²

(29)² = 20² + (21)²

841 = 400 + 441

841 = 841

HENCE VERIFIED.