Question

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

Answer 1

hb=4√3

Explanation:

Answer 2

AB = 8, HB = 6, Area of ∆ABC = 8
`√(3)`, Perimeter of ∆ABC = 12 + 4
`√(3)`

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
`AH^(2)+CH^(2)=AC^(2)`

then substitute, and get
`2^(2) + CH^(2) = 4^(2)`

calculate and get CH = 2
`√(3)`

then the height = CH = 2
`√(3)`

solve the area and get

Area of ∆ABC = 8
`√(3)`

(optional perimeter)

to find perimeter of ∆ABC:

you add AC + CB + AB

you find CB by using opposite to 30°

CB = CH*2 = 2
`√(3)`*2 = 4
`\sqrt3}`

so AC + CB + AB = 4 + 4
`\sqrt3}` + 8

Perimeter of ∆ABC = 12 + 4
`\sqrt3}`

Hope this helps!!