Question

Given: △DMN, DM=10 3 m∠M=75°, m∠N=45° Find: Perimeter of △DMN

Answer 1

The perimeter of
`\triangle DMN` will be 62.1939...

Explanation:

In
`\triangle DMN`, the length of side
`DM = 10√(3)` and the measures of
`\angle M` and
`\angle N` are 75° and 45° respectively.

As the sum of all angles in a triangle is always 180°, so the measure of
`\angle D` will be: 180°- (75°+45°) = 180°- 120° = 60°

Now using Sine rule, we will get......

`(MN)/(Sin(D))=(DN)/(Sin(M))=(DM)/(Sin(N))\\ \\ (MN)/(Sin(60))=(DN)/(Sin(75))=(10√(3))/(Sin(45))\\ \\ MN= (10√(3))/(Sin(45))*Sin(60)=21.2132...\\ \\ DN=(10√(3))/(Sin(45))*Sin(75)=23.6602...`

So, the perimeter of
`\triangle DMN` will be:
`DM+MN+DN = 10√(3)+21.2132...+23.6602... =62.1939...`