Question

right triangle ABC has area 32√3cm^2. The measure of <A = 30, m<B=90. What is the length of BC? AB? AC? Express all answers in simplest radical form.​

Answer 1

Answers:

BC = 8

AB = 8*sqrt(3)

AC = 16

"sqrt" stands for "square root"

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Explanation:

Angle C is 60 degrees because adding the angles gets A+B+C = 180. Put another way, C = 180-A-B. Therefore, C = 180-30-90 = 60.

We have a 30-60-90 triangle with

• BC as the short leg (since it is opposite the smallest angle A = 30)
• AB as the long leg (because it is opposite the medium angle C = 60)
• AC as the hypotenuse (it is opposite the largest angle, so AC is the longest side)

Check out the diagram below.

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From here, we use the 30-60-90 triangle template. The short leg is x, the hypotenuse is 2x and the long leg is x*sqrt(3). The area of this template triangle is

area = 0.5*base*height

area = 0.5*(short leg)*(long leg)

area = 0.5*x*(x*sqrt(3))

area = 0.5*x^2*sqrt(3)

set this equal to 32*sqrt(3) and you should find that 0.5x^2*sqrt(3) = 32*sqrt(3) solves to x = 8

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Therefore we can say,

• BC = short leg = x = 8
• AB = long leg = x*sqrt(3) = 8*sqrt(3)
• AC = hypotenuse = 2x = 16