Question

The altitude of a triangle is three-fifth the length of its corresponding base. If the altitude is increased by 4 cm and the base is decreased by 2 cm, then the area of the triangle remains the same. Find the base of the triangle in cm.​

Answer 1

Let x be the length of altitude of the triangle and y the length of the corresponding base.

Since altitude is one-fifth the length of the base, we have

x=5y3 (i)

Also, according to the question increasing altitude by 4 cm and decreasing base by 2 cm does not change the area of the triangle.

Since area of the triangle is equal to 12bh, we have

12xy=12(x+4)(y−2)⇒xy=(x+4)(y−2)

Expanding the expression on RHS, we get

xy=xy−2x+4y−8⇒2x=4y−8 ⇒x=2y−4 (ii)

Substituting the value of x from equation (i) in equation (ii), we get

5y3=2y−4

Adding 4 on both sides, we get

5y3+4=2y

Subtracting 5y3 from both sides, we get

5y3+4−5y3=2y−5y3⇒6y−5y3=4⇒y3=4

Multiplying both sides by 3, we get

3y3=12⇒y=12

Substituting the value of y in equation (ii), we get

x = 2(12)-4 = 24-4 = 20.

Hence the altitude of the triangle is 20cm, and the length of the base is 12 cm.