Question

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

Answer 1

To find the perimeter of △DMN, we utilize the fact that it is a 30-60-90 triangle with side length ratios of 1:√3:2. With DM given as 10√3, DN is 5√3, and MN is 10√3. The perimeter is therefore 25√3 units.

Step-by-step explanation:

To find the perimeter of △DMN, we need to determine the lengths of all three sides and sum them up. We are already given that DM = 10√3. Since the interior angles of a triangle add up to 180°, we can find the measure of ∠D by subtracting the measures of ∠M and ∠N from 180°:

m∠D = 180° - (m∠M + m∠N) = 180° - (75° + 45°) = 60°.

With angles M and D being 75° and 60° respectively, and angle N being 45°, we recognize that △DMN is a 30-60-90 triangle. This type of triangle has side lengths in the ratio of 1:√3:2. Since DM is the side opposite the 60° angle, it corresponds to the √3 part of the ratio. Therefore, DN (opposite the 30° angle) will be half of DM, and MN (the hypotenuse) will be twice DN.

DN = DM / 2 = (10√3) / 2 = 5√3

MN = 2 × DN = 2 × 5√3 = 10√3

The perimeter P of △DMN is then calculated as:

P = DM + DN + MN = 10√3 + 5√3 + 10√3 = 25√3

Thus, the perimeter of △DMN is 25√3 units.

Answer 2

Correct Answer is: Perimeter of ΔDMN is 54.873 units.

Solution:-

Given that DM=10√3,m∠M=75°,m∠N=45°

So,m∠D= 180-(75+45) = 60°

So to find perimeter we need other 2 sides also.

Let us use sine rule to find them.

`(DM)/(sin(N))=(DN)/(sin(M)) =(MN)/(sin(D))`

`(10√(3) )/(sin(45)) = (DN)/(sin(75))=(MN)/(sin(60))`

`DN=(10√(3) )/(sin(45))Xsin(75) = 23.66`

`MN=(10√(3) )/(sin(45)) Xsin(60)=21.213`

Hence perimeter = DM+MN+DN

= 10+21.213+23.66

=54.873 units