Consider the following diagram:
 

Suppose I told you that AB = 6, BC = 8, DE = 3, and EF = 4. Is that enough information to prove that the two triangles are similar?

It sure is! Because the Pythagorean Theorem shows us that AC = 10 and DF = 5, which gives us 3 pairs of sides that are in equal proportion.

So what is the ratio of a side of ABC to the corresponding side of DEF? The ratio is 2, because 6/3 = 8/4 = 10/5 = 2.

Now here's an interesting question: what is the ratio of the areas of those two triangles?

Well, if we remember that the area of a triangle is¹/₂bh, then we can easily find the two areas:

Area of ABC = ¹/₂(6)(8) = 24
Area of DEF = ¹/₂(3)(4) = 6.

Interesting; the ratio of the areas isn't 2 - it's 4! I wonder if we can find a pattern.

Suppose DE = 5, EF = 12, AB = 15, and BC = 36. These are still similar, because the hypotenuses of the triangles are 13 and 39, so the ratio of sides of ABC to corresponding sides of DEF is 3 (15/5 = 3)

What about the areas?

Well, the area of DEF is 30, and the area of ABC is 270.

What is the ratio of the areas here? It's 9.

This is interesting: when the ratio of the sides was 2, the ratio of the areas was 4, and when the ratio of the sides was 3, the ratio of the areas was 9. Does this look like a pattern?

2² = 4
3² = 9

It turns out that this pattern always works - if ratio of the sides of two similar triangles is x then the ratio of the areas of the triangles is x²

And they don't even have to be right triangles!

Example 1: Suppose ABC is similar to DEF, with AB = 5 and DE = 8. If the area of ABC is 50, what is the area of DEF?

Solution: The ratio of the sides is 5/8, so the ratio of the areas must be (5/8)² or 25/64. So if the area of ABC = 50, then the area of DEF is 50(64/25) = 128.

Example 2: If XYZ is similar to JKL, and the area of XYZ is 4 times the area of JKL, then how many times the length of JK is XY?

Solution: The ratio of the areas is the square of the ratio of the sides, so if the ratio of the areas is 4, the ratio of the sides must be the square root of 4, or 2.

1.

If the ratio of the corresponding sides of two similar triangles is 5, what is the ratio of their areas?

2.

If the ratio of the areas of two similar triangles is 4/9, what is the ratio of the corresponding sides?

3.

Triangle ABC is similar to triangle JKL. AB = 10 and JK = 15. If the area of ABC is 16, what is the area of JKL?

4.

If the ratio of the corresponding sides of two similar triangles is 1, what is the ratio of the areas?

5.

Triangle ABC is similar to triangle XYZ. If the area of ABC is 16 times the area of XYZ, and XY = 20, what is the length of AB?

6.

DEF is similar to GHI. If the area of DEF is one third the area of GHI, and GH = 10, what is the length of DE?

7.

Under what circumstances is the ratio of the areas of two triangles less than the ratio of the corresponding sides?

8.

The ratio of the areas of two similar triangles is 8x+1, and the ratio of their corresponding sides is x + 2. What is the value of x?

9.

ABC is similar to DEF. AB = x, DE = x - 1, the area of ABC = x - 2, area of DEF = x - 1. What is the value of x?

10.

ABC is similar to DEF and DEF is similar to GHI. AB is three times DE, and EF is 4 times HI. If the area of GHI is 10, what is the area of ABC?