Answer:
20.7 cm
Explanation:
Given the bow-tie shaped figure shown, you want the measure of side DE.
Analysis
The ratio of sides DC:AC is 16/24 = 2/3. The ratio of sides EC:BC is 19/31, which is not 2/3. This means the triangles are not similar.
In triangle ABC, two sides and an angle are given, which is sufficient information to solve that triangle. In triangle CDE, two sides are given, so we need at least one angle in order to solve that triangle.
The angle we can find is angle DCE, which is a vertical angle to ACB, so congruent to it.
With the sides given in ∆ABC, we can solve for angle B using the Law of Sines, then use the sum of angles in a triangle to find the interior angle at C.
Once we know interior angle C, we can use the Law of Cosines to solve for DE.
Law of Sines
sin(B)/AC = sin(A)/BC
sin(B) = AC/BC·sin(A) = 24/31·sin(64°) ≈ 0.695841
B ≈ 44.0942°
Then angle C is ...
C = 180° -64° -44.0942° = 71.9058°
Law of Cosines
DE² = CD² +CE² -2·CD·CE·cos(C)
DE² = 16² +19² -2·16·19·cos(71.9058°) ≈ 428.167
DE ≈ √428.167 ≈ 20.692
The length of DE is about 20.7 cm.
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Additional comment
The first attachment shows the figure drawn to scale.
The 2nd and 3rd attachments show triangle solver solutions to the triangles. The angle measure 71.906° was copied from the first solution to the second. In the above calculation, we used full calculator precision throughout, rounding at the end.