find the perimeter and area of an isosceles triangle whose vertex is 35 degrees and whose two equal sides each measure 15 cm

Answer 1

Hi William L

You should definitely draw your triangle. The Isosceles triangle you describe has two equal sides of 15 cm each with a vertex angle between them(the two equal sides) of 35 degrees, please let me know if this correct.

You can calculate the area as Area = 1/2*a*b*sin(ϒ)

Where a and b are legs and ϒ is the angle between them

Since a = b both sides can be labeled side a

Each base angle A = 72.5°; where the calculation of the base angle A = (180 - 35)/2 = 145/2 = 72.5

The formula above becomes Area = 1/2*a²*sin(ϒ)

Vertex angle = ϒ = 35° with side x

Substituting this is

Area = 1/2(a²)sin(35)

A= 1/2(15²)sin(35) = 64.53

Since you have an isosceles triangle, and you need to find the length of the side opposite the vertex you

You can use the Law of Sines or the Law of Cosines

Law of Sines

a/sin(base angle) = x/sin(vertex angle)

15/sin(72.5°) = x/sin(35°)

Solve for x

x = (15*sin(35°))/sin(72.5°) = 9.02 cm

Law of Cosines

x = SRT((2(15²) -( 2(15²)cos(35°))

x = SQRT(450 - (450*0.81915))

x = SQRT(450 - 368.618)

x = 9.02

The Perimeter is just the sum of all the sides for a Triangle so 2a + x will be the perimeter

Perimeter = 2a + x = 2(15) + 9.02 = 39 .02 cm

If you have not had Trigonometry yet, please draw your Isosceles Triangle, drop a height from the center of the vertex angle to the base to form a two equal Right Triangles and split the length of the base into two equal halves. In one of your Right Triangle, 15 cm will be the hypotenuse, the side opposite the base angle of 72.5° will be the height. Using the relationship between the sides and angles in a Right Triangle, you can find the height as follows:

sin(72.5°) = height/15 cm

Solve for the height

height = 15 cm * sin(72.5°) = 14.306 cm

You can use this height to calculate the missing side one of your right triangles, the shortest side

SQRT(15² - 14.306²) = 4.510

Since you formed two Equal Right Triangles, the shortest side of one only represents half total base for your Isosceles Triangle, so you need to double this

Base of your isosceles triangle is 2(4.51) = 9.02

You know the height and you can again calculate the Area as 1/2bh

Area = 1/2*9.02*14.306 = 64.53

Perimeter = 2(15) + 9.02 = 30 + 9.02 = 39.02

There are some other ways to approach the calculations using Right Triangles and other formula’s for the area including Heron’s formula once all the sides are known. I hope the above helps

Answer 2

See the video.