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

Question

Given: △DMN, DM=10

    m∠M=75°, m∠N=45°

    Find: Perimeter of △DMN

Answer 1

Law of sines:

sin 45 / 10 = sin 60/ x

x = sin 60 * 10 / sin 45 = sqrt(3)/2 * 10 * 2 / sqrt(2)

= sqrt(3) / 2 * 10 * sqrt(2)

= 5 * Sqrt(6)

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sin 45 / 10 = sin 75 / y

y = sin 75 * 10 / sin 45 = sin 75 * 10 * 2 / sqrt(2)

= sin 75 * 10 * sqrt(2)

= 13.066054....

the perimeter is 10 + 5*Sqrt(6) + 13.06605425... = 35.90770275.....

Answer 2

This problem is solved by applying using the Triangle Sum Theorem (the measures of the interior angles of a triangle sum to 180 degrees) to find the m∠D = 180 - 75 - 45 = 60, then use the Law of Sines to find the missing sides.

The Law of Sines states that the ratios of the sides to the sine of the angle opposite for any triangle is a constant (all can be set equal to each other). In terms of the triangle shown for the question, the Law of Sines gives:

DM ⁄ sin N = MN ⁄ sin D = ND ⁄ sin M

so:

10 ⁄ sin 45 = MN ⁄ sin 60 ==> MN = 10 *[(sin 60) ⁄ (sin 45)] = 12.2

10 ⁄ sin 45 = ND ⁄ sin 75 ==> ND = 10 *[(sin 75) ⁄ (sin 45)] = 13.7

The perimeter of any shape is the distance around the outside (a.k.a. the sum of the sides).

P = DM + MN + ND = 10 + 12.2 + 13.7 = 35.9