NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 5

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 5

Question:

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Given:

Type of triangle: Right triangle.

Altitude = Base - 7 cm.

Hypotenuse = 13 cm.

To Find:

The lengths of the base and altitude.

Formula:

WKT, for a right triangle, by Pythagorean Theorem: Base2 + Altitude2 = Hypotenuse2.

WKT, For a quadratic equation ax² + bx + c = 0, we can find two numbers m and n such that:

                        ax² + bx + c = ax² + mx + nx + c

where

  • m × n = a × c
  • m + n = b

Cases:

  • If (a × c > 0) and b < 0, both m and n are negative.
  • If (a × c > 0) and b > 0, both m and n are positive.
  • If (a × c < 0), the numbers m and n have opposite signs.

Solution:

Let the base of the right triangle (in cm) = x
According to the given condition, the altitude is 7 cm less than its base.

⇒ Altitude (in cm) = x - 7

The hypotenuse is given as 13 cm.
By Pythagorean Theorem: Base2 + Altitude2 = Hypotenuse2

⇒ x2 + (x - 7)2 = 132
Expand (x - 7)2:

WKT, (a - b)2 = a2 - 2ab + b2.

⇒ x2 + (x2 - 14x + 49) = 169
Combine like terms:

⇒ 2x2 - 14x + 49 = 169
Rearrange the equation into the standard quadratic form ax2 + bx + c = 0:

⇒ 2x2 - 14x + 49 - 169 = 0

⇒ 2x2 - 14x - 120 = 0
Divide the entire equation by 2 to simplify:

⇒ x2 - 7x - 60 = 0
Now, we solve this quadratic equation by factorization.

Here, a = 1, b = -7, c = -60.
Product (a × c) = 1 × (-60) = -60.

Sum (b) = -7.
We need to find two numbers whose product is -60 and sum is -7.

Take the factors of the product without sign.

WKT, 60 = 12 x 5

The two numbers we want are m = 12 and n = 5.

Since product is -60 and -60 < 0 both numbers have opposite signs.

Case 1: Let m = 12 and n = -5; product = -60 but sum = 7, therefore the two numbers can't be 12 and -5.

Case 2: Let m = -12 and n = 5; product = -60 and sum = -7, therefore the two numbers are -12 and 5.

Rewrite the middle term (-7x) using these numbers:

⇒ x2 - 12x + 5x - 60 = 0
Factor by grouping:

⇒ x(x - 12) + 5(x - 12) = 0
Factor out the common binomial (x - 12):

⇒ (x + 5)(x - 12) = 0
Set each factor equal to zero to find the values of x:
Case 1: x + 5 = 0

⇒ x = -5
Case 2: x - 12 = 0

⇒ x = 12
Since the length of a side cannot be negative, we take x = 12 cm.

Therefore, Base = 12 cm.

Altitude = x - 7 = 12 - 7 = 5 cm.

Result:

The base of the right triangle is 12 cm and the altitude is 5 cm.
© Kaliyuga Ekalavya. All rights reserved.

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