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## Solved Examples

**Question 1:**The base of a triangle is 2 m more than twice its altitude. If the area of the triangle is 30 square m. Find its altitude.

**Solution:**

Let the altitude of the triangle = x

therefore, base of the triangle = 2 + 2x

Given, area of the triangle = 30

Area of the triangle = $\frac{1}{2}$ Base * Height

=> 30 = $\frac{1}{2}$ * (2 + 2x) * x

=> 30 * 2 = (2 + 2x) * x

=> 60 = 2x + 2x

=> 2x

Solve for x, 2x

=> 2x

=> 2x(x + 6) - 10(x + 6) = 0

=> (2x - 10)(x + 6) = 0

either 2x - 10 = 0 or x + 6 = 0

=> 2x = 10 or x = - 6

Length must be positive, so neglect x = - 6.

=> 2x = 10

=> x = 5

Hence altitude of the triangle is 5 m

therefore, base of the triangle = 2 + 2x

Given, area of the triangle = 30

Step 1:Step 1:

Area of the triangle = $\frac{1}{2}$ Base * Height

=> 30 = $\frac{1}{2}$ * (2 + 2x) * x

=> 30 * 2 = (2 + 2x) * x

=> 60 = 2x + 2x

^{2}=> 2x

^{2}+ 2x - 60 = 0

Step 2:Step 2:

Solve for x, 2x

^{2}+ 2x - 60 = 0=> 2x

^{2}+ 12x - 10x - 60 = 0=> 2x(x + 6) - 10(x + 6) = 0

=> (2x - 10)(x + 6) = 0

**Step 3:**either 2x - 10 = 0 or x + 6 = 0

=> 2x = 10 or x = - 6

Length must be positive, so neglect x = - 6.

=> 2x = 10

=> x = 5

Hence altitude of the triangle is 5 m

**. answer****Question 2:**The sum of the ages of a father and his son is 45. 5 years ago, father's age is three years more than thrice the sons age. Find the age of the son 4 years hence.

**Solution:**

Let the age of the son = x years

Therefore the age of the father = 45 - x

(Sum of their ages is 45)

Step 1:

5 years ago:

Son's age = x - 5

and father's age = 45 - x - 5 = 40 - x

Step 2:

The problem states:

Father's age = 3 + 3(Son's age)

=> 40 - x = 3 + 3(x - 5)

=> 40 - x = 3 + 3x - 15

=> 40 - x = 3x - 12

=> 3x + x = 40 + 12

=> 4x = 52

Divide each side by 4

=> x = $\frac{52}{4}$

=> x = 13

Son's present age = 13 years

=> After 4 years son will be of = 13 + 4 = 17 years.

Therefore the age of the father = 45 - x

(Sum of their ages is 45)

Step 1:

5 years ago:

Son's age = x - 5

and father's age = 45 - x - 5 = 40 - x

Step 2:

The problem states:

Father's age = 3 + 3(Son's age)

=> 40 - x = 3 + 3(x - 5)

=> 40 - x = 3 + 3x - 15

=> 40 - x = 3x - 12

=> 3x + x = 40 + 12

=> 4x = 52

Divide each side by 4

=> x = $\frac{52}{4}$

=> x = 13

Son's present age = 13 years

=> After 4 years son will be of = 13 + 4 = 17 years.