9514 1404 393
Answer:
BD < DC < BC < AC < AB
Step-by-step explanation:
The law of sines can be used to figure this. First we need to know the unmarked angles.
∠ABC = 180° -50° -67° = 63°
∠BCD = 180° -57° -72° = 51°
The law of sines tells us the ratio of side lengths is equal to the ratio of opposite angles. Since side BC is common to both triangles, we can let it have a measure of 1. Then the lengths we develop for the other sides represent their ratio to the length of BC.
AB/sin(C) = BC/sin(A) ⇒ AB = sin(C)/sin(A) = sin(67°)/sin(50°) ≈ 1.202
BD/sin(C) = BC/sin(D) ⇒ BD = sin(C)/sin(D) = sin(51°)/sin(72°) ≈ 0.817
AC/sin(B) = BC/sin(A) ⇒ AC = sin(B)/sin(A) = sin(63°)/sin(50°) ≈ 1.163
BC = 1 . . . . by definition above
DC/sin(B) = BC/sin(D) ⇒ DC = sin(B)/sin(D) = sin(57°)/sin(72°) ≈ 0.882
The relative lengths from least to greatest are ...
0.817 (BD), 0.882 (DC), 1.000 (BC), 1.163 (AC), 1.202 (AB)
Then the order of sides is ...
BD < DC < BC < AC < AB
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The triangle solver results attached confirm these values. In the second attachment, node D is represented by node A.
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Additional comment
Angles B and C are ambiguous angle designators, as there is one such angle in each of the two triangles. We trust you can figure which one is meant by considering the angle measure and the other angle in the proportion. The law of sines only pertains to angles that are inside the same triangle.
