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Q.

The altitude of a triangle is five-thirds the length of the corresponding base. If the altitude is increased by 4cm and the base is decreased by 2cm, the area of the triangle remains the same. Find the altitude and the base of the triangle:

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a

14cm, 21cm

b

20cm, 12cm

c

18cm, 24cm

d

17cm, 26cm

answer is B.

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Detailed Solution

Concept: Use the statement in the question to form two linear equations in two variables x and y. Solving the formulated linear equations using methods - substitution, elimination, graphically or using matrices.
Given: the altitude of a triangle is five thirds the length of the corresponding base, altitude increased by 4 cm and the base is decreased by 2 cm , the area remains same,
Now, Suppose the length of altitude of the triangle is x and the length of the base is y,

x = \frac{5 y}{3}

Altitude increased by 4cm and the base is decreased by 2cm,
The area of triangle is

\frac{1}{2} bh

,

\frac{1}{2} xy = \frac{1}{2} (x + 4) (y - 2)

xy = (x + 4) (y - 2)

xy = xy - 2 x + 4 y - 8

2 x = 4 y - 8

\frac{5 y}{3} = 2 y - 4

Adding 4 both sides,

\frac{5 y}{3} + 4 = 2 y,

Subtracting

\frac{5 y}{3}

on both sides,

\frac{5 y}{3} + 4 - \frac{5 y}{3} = 2 y - \frac{5 y}{3}

\frac{6 y - 5 y}{3} = 4

\frac{y}{3} = 4

y = 12

Now, substituting value of y

x = 2 (12) - 4

x = 20

Altitude of triangle is 20 cm,
Length of base is 12 cm,
Hence, the correct option is 2.

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