Final answer:
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.