if the altitude and the base of a triangle are each increased by 4 inches the area of the triangle will be increased by 42 inches if the altitude is increased

Question

if the altitude and the base of a triangle are each increased by 4 inches the area of the triangle will be increased by 42 inches if the altitude is increased by 3 inches and the base is decreased by 2 inches the area will be increased by 5 inches.Find the base and altitude of the triangle

Answer 1

For the area of a triangle we have A=ab/2 where A represents the area, a represents the altitude, and b represents the base.

If the area is increased by 42 square inches when the altitude and base are both increased by 4 inches, we have A + 42 =(a+4)(b+4)/2.

If the area is increased by 5 square inches when the altitude is increased by 3 inches and the base decreased by 2 inches, we have A+5=(a+3)(b-2)/2.

Simplifying gives us 2A=ab, 2A + 84 = ab + 4a + 4b + 16 (which further simplifies to 2A + 68 = ab + 4a + 4b), and 2A + 10 = ab - 2a + 3b - 6 (which further simplifies to 2A + 16 = ab -2a + 3b).

Since 2A = ab, we can substitute and get 68 = 4a + 4b (simplifying to 17 = a + b) and 16 = -2a + 3b.

We can rewrite this system as a + b = 17 and -2a + 3b = 16.

Multiplying the first equation by 2 gives us 2a + 2b = 34 and -2a + 3b = 16. We can add these equations to eliminate a and get 5b = 50, or b = 10.

Via substitution into a + b = 17, we get a = 7.

Therefore, the base (b) is 10 inches long and the altitude (a) is 7 inches long.