A triangle with an area of 728 square feet has a height that is 8 less than 8 times its base. Find the base and height, in feet, of the triangle.

Answer 1

Area of triangle = 1/2 x height x base

in this case, 728 = 1/2 hb => hb = 1456

but we are also told that h = 8b - 8 (height is 8 less than 8 times the base)

hence we can substitute to eliminate h, and get (8b - 8)(b) = 1456

At this point we can EITHER use trial-and-error (guess!)...

try b = 10 for 72 x 10 = 720 too small

try b = 15 for 112 x 15 = too big

try b = 14 for 104 x 14 = 1456 just right :-) and it has given us the height too.

OR factorize...

(8b - 8)(b) = 1456

=> 8b² - 8b - 1456 = 0

=> 8 (b² - b - 182) = 0

=> 8 (b + 13)(b - 14)

Hence either b + 13 = 0 => b = -13 (negative base not possible for triangle)

or b - 14 = 0 => b = 14

If b = 14 substitute to get h in h = 8b - 8 = 8(14) - 8 = 104

Hence, base = 14 feet and height = 104 feet (or vice versa)

Double-check: 1/2 x 14 x 104 = 7 x 104 = 700 + 28 = 728 which is what we need.

Answer 2

The area of a triangle is (1/2)*b*h where b is base and h is height.

So our first equation is (1/2)*b*h = 728.

If the height is 8 less than 8 times its base, then h = 8*b - 8.

If we substitute 8*b - 8 for h in the 1st equation:

(1/2)*b*(8*b-8) = 728

(1/2)*(8*b²-8b) = 728

4b² - 4b = 728

b² - b = 182

Since we have a quadratic, lets set the set the equation equal to 0 and factor:

b² - b - 182 = 0

(b + 13)(b - 14) = 0

We now have 2 possibilities for the length of the base:

1. b + 13 = 0 -> b = -13
2. b - 14 = 0 -> b = 14

Since the base cannot have a negative length, it must have a length of 14.

We can now solve for h:

h = 8*b - 8

h = 8*14 - 8

h = 104

The base is 14 feet.

The height is 104 feet.

Answer 3

H = 8B - 8

Area = 728 = (1/2)BH = (1/2) B(8B-8)

1456 = 8B^2 - 8B

B^2 - B = 1456/8 = 182

complete the square, add to both sides 1/2 the coefficient of B all squared =1/4

B^2 - B + 1/4 = 1/4 + 182

(B-1/2)^2 = 729/4

B-1/2 = 27/2

B = 27/2+1/2 = 28/2 = 14 feet base

H = 8B-8 = 8(14)-8= 104 feet high

A=(1/2)BH = (1/2)(14)(104) = 7(104) = 728 sq. ft.