Given isosceles triangle ABC with base BCeach base angle measures 75 deg and each leg is 30 inches longfind the area of this triangle 225sqrt(2); 15/2 225 225sqrt(3) 450 None of these

Question

Given isosceles triangle ABC with base BCeach base angle measures 75 deg and each leg is 30 inches longfind the area of this triangle 225sqrt(2); 15/2 225 225sqrt(3) 450 None of these

Answer 1

Let b = base and h = height

sin75° = h/30 So, h = 30sin75°

cos75° = (b/2) / 30 So, b = 60cos75°

Area = (1/2)bh = 900sin75°cos75° = 450(2sin75°cos75°) = 450sin150° = 450(1/2) = 225

Answer 2

2 area formulas you could use, easiest is:

Area=.(1/2)bcsinA

= one half side times side times sine of the angle between the two sides (but if you use 30 is a leg opposite 75 degrees, then Area = 225 in^2)

=.5(30)(30)sin75

= 450sin75

=about 450(.96595926) Correct Answer: None of the Above

= 434.66662 in^2

or easiest way is go online and use a triangle calculator. They do the work for you and there's several free online calculators, all give 434.666

or use the more standard formula, but it's easy to make a mistake calculating the height and for a given base.

Area = bh/2 = one half base times height

= about .5(36.525)(23.8006)

= 434.67 square inches

side opposite 75 degrees is the longest side, greater than 30 inches

other 2 angles = (180-75)/2 = 105/2 = 52.5 degrees

by Law of Sines a/sin75 = 30/sin52.5 = ratio of side to sine of angle opposite that side

a=30sin75/sin52.5 = about 36.526 inches

divide the triangle in half, down the middle and find the height

angle opposite the height, h, = 52.5 degrees side adjacent to 52.5 degrees = b/2 = 36.526/2 = 18.263 inches

height = (b/2)tan52.5 = 18.263tan52.5 = 23.8008

sin75 = sin(30+45) = sin30cos45 + sin45cos30 = .5(sqr2)/2 + (sqr2)/2)(sqr3)/2 = sqr2/4 +sqr6/4

= (sqr2 +sqr6)/4= about .965926

cos75 = cos(30+45) = cos30cos45 - sin30sin45 = (sqr3)/2)(sqr2)/2 - (1/2)(sqr2)/2 = (sqr6)/4 - (sqr2)/4

= (sqr6 -sqr2)/4 = about .25882

while 900sin75cos75 = 900(.965926)(.25882) = 225, that's Not the Area, as it may confuse h for hypotenuse with h for height. but if the triangle has 30 opposite 75 degrees, 225 is correct

30sin75 = about 28.9777 = the height, but from a different side/base

60cos75 = about 15.529 does not = base (unless 30 is opposite 75), the base=30 otherwise

bh/2 = about (30)28.977/2 = 434.67