In right triangle A B C , angle C is the right angle and B C = 162 . Point

Question

In right triangle A B C , angle C is the right angle and B C = 162 . Point D on side A B is connected by a line segment with point E on side A C such that line segment D E is parallel to side B C and C E = 2 A E . What is the length of line segment D E ?

Accepted Answer

The correct answer is 54 . It&rsquo;s given that in triangle A B C , point D on side A B is connected by a line segment with point E on side A C such that line segment D E is parallel to side B C . It follows that parallel segments D E and B C are intersected by sides A B and A C . If two parallel segments are intersected by a third segment, corresponding angles are congruent. Thus, corresponding angles C and A E D are congruent and corresponding angles B and A D E are congruent. Since triangle A D E has two angles that are each congruent to an angle in triangle A B C , triangle A D E is similar to triangle A B C by the angle-angle similarity postulate, where side D E corresponds to side B C , and side A E corresponds to side A C . Since the lengths of corresponding sides in similar triangles are proportional, it follows that D E B C = A E A C . Since point E lies on side A C , A E + C E = A C . It's given that C E = 2 A E . Substituting 2 A E for C E in the equation A E + C E = A C yields A E + 2 A E = A C , or 3 A E = A C . It&rsquo;s given that B C = 162 . Substituting 162 for B C and 3 A E for A C in the equation D E B C = A E A C yields D E 162 = A E 3 A E , or D E 162 = 1 3 . Multiplying both sides of this equation by 162 yields D E = 54 . Thus, the length of line segment D E is 54 .

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