Question

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

Answer 1

Answer:

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

Step-by-step explanation:

In $\triangle DMN$, the length of side $DM = 10\sqrt{3}$ 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......

$\frac{MN}{Sin(D)}=\frac{DN}{Sin(M)}=\frac{DM}{Sin(N)}\\ \\ \frac{MN}{Sin(60)}=\frac{DN}{Sin(75)}=\frac{10\sqrt{3}}{Sin(45)}\\ \\ MN= \frac{10\sqrt{3}}{Sin(45)}*Sin(60)=21.2132...\\ \\ DN=\frac{10\sqrt{3}}{Sin(45)}*Sin(75)=23.6602...$

So, the perimeter of $\triangle DMN$ will be: $DM+MN+DN = 10\sqrt{3}+21.2132...+23.6602... =62.1939...$