A triangle with area 21 square inches has a height that is four less than six times the base. Find the base and height of the triangle (in inches).

Question

Solve by using methods of factoring, the square root principle, or the quadratic formula. Round your answers to the nearest tenth.

Answer 1

Area of a triangle is 1/2bh

In this case h = 6b - 4

So A = 21 = 1/2b(6b - 4)

21 = 1/2b(6b - 4)

21 = 3b² - 2b

3b² - 2b - 21 = 0

Solve by factoring:

3b² - 2b - 21 = 0

(3b + 7)(b - 3) = 0

Setting each binomial equal to zero:

3b + 7 = 0 meaning that b = -7/3

b - 3 = 0 meaning b = 3

Solve by using the square root principle:

3b² - 2b - 21 = 0

3b² - 2b = 21

3(b² - 2/3b) = 21

b² - 2/3b = 7

b² - 2/3b + 1/9 = 7 + 1/9

(b - 1/3)² = 63/9 + 1/9

(b - 1/3)² = 64/9

b - 1/3 = ±√(64/9)

b - 1/3 = ±8/3

Breaking this into two parts:

b - 1/3 = 8/3 meaning b = 3 and b - 1/3 = -8/3 meaning b = -7/3

Solve using the quadratic formula:

3b² - 2b - 21 = 0 or 3x² - 2x - 21 = 0

The quadratic formula says for ax² + bx + c = 0 then x = [-b ± √(b² - 4ac)]/(2a)

In this case, a = 3, b = -2, and c = -21 so:

x = [-(-2) ± √((-2)² - 4(3)(-21))]/(2•3)

x = [2 ± √(4 + 252)]/6

x = (2 ± √256)/6

x = (2 ± 16)/6

Breaking this into two parts:

x = (2 + 16)/6 meaning x = 3 and (2 - 16)/6 meaning x = -7/3 (or b = 3 or -7/3)

In these answers, b = -7/3 does not make sense since the base cannot be a negative answer, so the only viable answer is b = 3. Therefore the base is 3 and the height is 14.

Answer 2