Question

the height of a triangle is 5/3rd the length of its corresponding base. if the height is increased by 4 cm and the base is decreased by 2 cm. the area of the traingle remains the same. find the base. height and area of the triangle​

Answer 1

Given:-

• height of 5/3 rd of the length
• height is increased by 4cm
• base is decreased by 2cm

To find:-

• base and height of the triangle

Assumption:-

• Let the length of base of 1st triangle be x
• height of first triangle becomes 5/3rd of x = 5/3×x = 5/3x
• Base of 2nd triangle becomes x - 2
• Height of 2nd trianlge becomes 5/3x+4

Solution:-

1st case,

Base of 1st traingle = x cm

Height of 1st triangle = 5/3x cm

Area of 1st triangle = 1/2 × base × height

= 1/2 × x × 5/3x

= 5/2 x² cm²

2nd case,

Base of 2nd triangle is decreased by 2 cm = (x - 2)cm

Height of 2nd triangle is incresed by 4 cm = (5/3x + 4) cm

Area of 2nd triangle:-

$\frac{1}{2} \times {(x - 2)} \times ( \frac{5x}{3} + 4) \\ = x( \frac{5x}{3} + 4) - 2( \frac{5x}{3} + 4) \\ = \frac{5x {}^{2} }{3} + 4x - \frac{10x}{3} - 8 \\ = (\frac{5 {x}^{2} }{3} + 4x - \frac{10x}{3} - 8) cm {}^{2}$

According to the question,

Area of 1st triangle = Area of second triangle

$\frac{5 {x}^{2} }{3} = \frac{5 {x}^{2} }{3} + 4 x- \frac{10x}{3} - 8\\ = \frac{5x {}^{2} }{3} - \frac{5 {x}^{2} }{3} = 4x - \frac{10x}{3} - 8 \\ = > 8 = \frac{12x - 10x}{3} \\ = > 24 = 2x \\ = > x = \frac{24}{2} \\ = > x = 12$

To find Height and base:-

Base = x = 12cm

Height = 5/3x = 5/3 × 12 = 5×4 = 20cm

Now to find the area of the triangle:-

Area = 1/2 × base × height

= 1/2 × 12 × 20

= 12×10 cm²

= 120 cm²