icse class 10 question from quadratic equations , problem on geometrical figures . I want full answer please help ✨
Answers
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