Question

Given: ΔABC, AB=12, m∠A=60°,m∠C=45°, AB =8 Find the perimeter of ΔABC

Answer 1

Using the figure above, if AB = RT and ∠R = 70°, then ________.

ΔABC ~ ΔRTS

m ∠ S = 50°

ΔABC ≅ ΔRTS

m ∠ S = 60°

Explanation:

Answer 2

The perimeter of ΔABC is 43 units

Explanation:

An illustrative diagram is shown in the attachment below.

To find the perimeter of the triangle, we will first determine the length of the unknown sides.

First, we can determine /BC/ using the Sine rule

From Sine rule

`(SinA)/(a) = (SinB)/(b) = (SinC)/(c)`

∴
`(SinA)/(a) = (SinC)/(c)`

In the diagram,
`a = /BC/` and
`c = /AB/ = 12`

m∠A=60° and m∠C=45°

∴
`(Sin60)/(/BC/) = (Sin45)/(12)`

`/BC/ = (12 * Sin60)/(Sin45)`

`/BC/ = 6√(6)`

To find /AC/, will first determine m∠B

m∠A + m∠B + m∠C = 180° (Sum of angles in a triangle)

60° +m∠B + 45° = 180°

m∠B + 105° = 180°

m∠B = 180° - 105°

m∠B = 75°

Also, using the sine rule

`(SinB)/(b) = (SinC)/(c)`

From the diagram,
`b = /AC/`

`(Sin75)/(/AC/) = (Sin45)/(12)`

`/AC/ = (12 * Sin75)/(Sin45)`

`/AC/ = 6 + 6√(3)`

Now,

The perimeter of ΔABC = /AB/ + /BC/ + /AC/

=
`12 + 6√(6) + 6+6√(3)`

= 43.09 units

≅ 43 units

Hence, the perimeter of ΔABC is 43 units.