A billboard in the shape of a triangle has sides that measure 10ft, 8.5ft, and 6ft. Find the area of the triangle

Question

use Herons formula to find the area

Answer 1

Billboard:

a=6

b=8.5

c=10

α=acos((b^2+c^2-a^2)/(2bc))

α= " ((72.25+100-36)/(170))=.801acos=36.7292.....

a/sinα=b/sinβ=c/sinγ

6/36.729°sin=10.032....=8.5/.847: asin=57.91°

180-36.73-57.91=85.36=γ

Made into 2 rt triangles:

Need side "a".

a/sin36.7=b/sin53.3=8.5 (sin of rt angle=1)

.598*8.5=5.08=α

Area=Lh/2=10*5.08/2=25.416sq/ft

Answer 2

use Heron's formula

p=half the perimeter = (10+6+8.5/)/2 = 24.5/2 = 12.25

Area = sqr(p)(p-10)(p-8.5)(p-6) = 25.42

which is slightly less than the area of a right triangle with similar sides

or if you want find the angles and calculate the altitude of the triangle

use the law of cosines to calculate the area opposite the longest side

10^2 = 8.5^2 + 6^2 - 2(8.5)(6)cosA

100 = 72.25 + 36 - 112cosA

112cosA = 98.25 = 1.75 (it's almost a right triangle with area almost 26.5)

cosA = 1.75/112 = 0.015623

A =89.1 degrees

sinA= .9999

sinA/10 = sinB/8.5

sinB = 8.5(.9999)/10 = ,.849896

B = 58.2 degrees

C = 180-58.2-89.1 = 32.7 degrees