# Similar Triangle Length Problem

A triangle has sides $$3$$, $$4$$, and $$5$$. A second triangle has a perimeter of $$20$$. If the two triangles are similar, what is the length of the longest side of the second triangle?

The answer I got was $$25/3$$, but my teacher told me that was wrong. How do I get the right answer?

• I believe that you got the correct answer, unless you have mis-stated the question. – Mohammad Zuhair Khan May 5 at 17:59
• Check that the problem statement is correct, because if so, your answer is also correct. What did your teacher believe the answer should be? – Deepak May 5 at 18:29

## 1 Answer

Since the two triangles are similar, we know that $$a'=3\cdot k\qquad b'=4\cdot k\qquad c'=5\cdot k$$ Furthermore $$a'+b'+c'=(3+4+5)\cdot k=12k=20\iff k=\frac{20}{12}=\frac53$$ Therefore

$$c'=5\cdot\frac53=\frac{25}3$$

So your answer is right!

• Nice approach. +1. It's acceptable to just take the ratio of the perimeters immediately since the proportionality of perimeters can be assumed in similar figures, but your approach may be considered more rigorous. – Deepak May 5 at 18:30