In right ∆ABC with m∠B=30°, AC = 4.
Find AB, HB and Area of triangle ABC

AB = 8, HB = 6, Area of ∆ABC = 8, Perimeter of ∆ABC = 12 + 4

Step-by-step explanation:

To find AB:

∆ABC is an 30,60,90∆

Using the theorem, you can find AB = 2AC = 2*4 = 8

AB = 8

To find HB:

You need to find AH to subtract from AB

Construct CH, a perpendicular bisector to side AB

From before you can put together that m∠CAB = 60°

∆ACH is an 30,60,90∆

Using this method again, AH = AC/2 = 4/2 = 2

Then you subtract AH from AB = 8-2 = 6

HB = 6

To find the area of ∆ABC:

You use the (base*height)/2 method

base = AB = 8

to find the height, CH, you need to use the Pytha Theorem

and get

then substitute, and get

calculate and get CH = 2

then the height = CH =  2

solve the area and get

Area of ∆ABC = 8

(optional perimeter)

to find perimeter of ∆ABC:

you add AC + CB + AB

you find CB by using opposite to 30°

CB = CH*2 = 2*2 = 4

so AC + CB + AB = 4 + 4 + 8

Perimeter of ∆ABC = 12 + 4

Hope this helps!!

beanburito

step-by-step explanation: