Triangles DEF and HJK are similar. DEF has side lengths = 4, = 6, = 8. Which of the following could be the side lengths of HJK?

A. 2, 6, 7

B. 8, 12, 16

C. 5, 7, 9

D. 2, 5, 7

Answer. Option B: 8,12,16

Solution:

Like the triangles are similar, their sides must be proportional, then if:

DE=4, EF=6 and FD=8; and the proportional sides are HJ, JK and KH respectively, then:

HJ/DE=JK/EF=KH/FD

Replacing DE=4, EF=6 and FD=8 in the equation above:

HJ/4=JK/6=KH/8

In option A or D, if HJ=2 (the smaller side)

2/4=JK/6=KH/8

Simplifying the fraction

1/2=JK/6=KH/8

Using the first equality:

1/2=JK/6

Solving for JK: Multiplying both sides of the equation by 6:

6 (1/2) = 6 (JK/6)

6/2=6 JK/6

3=JK

JK=3

Then options A (JK=6) or D (JK=5) are no possibles.

Using HJ=8 (the smaller side) in the option B:

8/4=JK/6=KH/8

2=JK/6=KH/8

Using the first equality:

2=JK/6

Solving for JK: Multiplying both sides of the equation by 6:

6 (2) = 6 (JK/6)

12=6 JK/6

12=JK

JK=12

Using the second equality:

2=KH/8

Solving for KH: Multiplying both sides of the equation by 8:

8 (2) = 8 (KH/8)

16=8 KH/8

16=KH

KH=16

Option B is possible.

In option C, if HJ=5 (the smaller side)

5/4=JK/6=KH/8

Using the first equality:

5/4=JK/6

Solving for JK: Multiplying both sides of the equation by 6:

6 (5/4) = 6 (JK/6)

30/4=6 JK/6

15/2=JK

JK=15/2=7.5

Then option B (JK=7) is no possible.