A triangle with an area of 280 square yards has a height that is 6 more than 8 times its base. Find the base and height, in yards, of the triangle.

Answer 1

Let

h be the height of the triangle,

b be the base of the triangle,

A be the area of the triangle.

The formula for area of a triangle is

A = bh/2 (1)

The statement of the problem in translation from English into math is

A = 280 (2)

h = 6 + 8b (3)

Combining (1), (2) and(3):

280 = b(6 + 8b)/2

Transforming into standard quadratic equation format

4b² + 3b - 280 = 0

The solution of the quadratic equation is

b = (-3 + sqrt(9 + 4.4.280))/8 = 8

Plugging this value of b into (3):

h = 6 + 8.8 = 70

Answer 2

base*height/2=area

(b*((8b+6))/2=280

(8b^2+6b)/2=

4b^2+3b=

4b^2=

b=8

h=70

Answer 3

You need a bit of algebra, and the formula for Area of triangle A = 1/2 (H x B) where H=height, B=base

We therefore know 280 = 1/2 (HB) => HB = 560

And we know H in terms of B, as H = 8B + 6 ("a height that is 6 more than 8 times its base")

Which allows us to substitute for H in HB = 560 to get

(8B + 6)B = 560

=>8B² + 6B = 560

=>8B² + 6B - 560 = 0

=> 2(4B² + 3B - 280) = 0 (common factor)

=> 2(4B + 35)(B - 8) = 0 (factorizing)

so we have 4B + 35 = 0 => 4B = -35 => B = -35/4 (negative base not possible for triangle)

and B - 8 = 0 => B = 8

then substitute B to get H in equation H = 8(8) + 6 = 70

So our Base is 8 yards, and our Height is 70 yards.

(Or, of course, it could be Base of 70, with Height of 8.)

Double-check: 1/2 x 8 x 70 = 280 which is what we want.

Note: the factorizing above may be a little tricky for some, but there are factor calculators out there to help you. Alternatively, I think we can, instead, do a bit of trial and error starting with:

(8B + 6)B = 560

try base = 5 so 46 x 5 = 230 too small

try 10 so 86 x 10 = 860 too big

try 7 so 62 x 7 = 434 too small

try 8 so 70 x 8 = 560 just right :-) and tells us height too.