A triangle has a base of 8 inches and sides of 4 and 7 inches.

Question

What is the height of the triangle?

Round to the nearest hundredth.

 

The height is approximately __?__  in.

Answer 1

height = 3 1/2 inches

a triangle with base = 8 and sides 4 and 7 have a height of 3.50 inches

use the Law of Cosines

b^2 = a^2 + c^2 -2acCosB

8^2 = 4^2 + 7^2 - 2(7)(4)CosB where B is the angle opposite the base, b = 8

64 = 16 + 49 - 56CosB

CosB = (64-16-49)/(-56) = 1/56

B = Cos^-1(1/56) = 88.9768... = about 89 degrees. It's almost a right triangle with 8 as the hypotenuse

as a right triangle let 7 = base and 4 as the height, then area = 7(4)/2 = 14 in^2

with 8 = base, the height = h where 8(h)/2 = 14, 4h=14, h= 14/4 = 7/2 = 3.5 inches

but that's a close approximation

attempting closer approximation with B=88.9768, doesn't get any closer,

with the Law of Sines

sin88.9768/8 = sinC/7 = .99984055/8= .12498 = sinC/7, sinC = 7(.12498) = .87486804780

C = sin^-1(.87486478) = 61.02846756 = about 61 degrees

A = 180-88.9768 -61.02846756 = 29.99473244= about 180-89-61 = 30 degrees

with base = 8 and 30 degree angle with side 7, height = 7(1/2) = 3.5 inches

7sin29.99473244 = 3.499442651 = 3.499 rounded to 3 one thousandths or 3.50 to closest hundredths

area = hb/2 = 3.5(8)/2 = 14 in^2