common side of two triangles

Question

Two triangles share a common side. The lengths of the sides of the first triangle are in the ratio 5:6:7, while those of the second are in the ratio 4:9:11. If the lengths of all six sides are integers, what is the smallest possible length of this common side. 

Answer 1

5 + 6 + 7 = 18

4 + 9 + 11 = 24

 

What is the least common multiple of 18 and 24?

Use that number to determine the length of each of the six sides.

Answer 2

By using proportions, 5:6:7 has a similar ratio to 10:12:14. 4:9:11 has a similar ratio to 12:27:33. Notice the ratios 10:12:14 and 12:27:33 both share the number 12. Another strategy other than proportions is finding the least common multiple (LCM) between one side of the 1st triangle and one side of the 2nd triangle. LCM(x,y) where x is either 5,6, or 7 and y is either 4,9, or 11. We have 9 pairs: LCM(5,4)=20, LCM(5,9)=45, LCM(5,11)=55, LCM(6,4)=12, LCM(6,9)=18, LCM(6,11)=66, LCM(7,4)=28, LCM(7,9)=63, LCM(7,11)=77. The pair with the smallest LCM is 12. Therefore based on these two alternative ways, the smallest possible length of the common side is 12.