"Triangle DEF is similar to triangle ABC, and the length of side DE is 6 cm. The length of side AB is 2 cm. If the area of triangle ABC is 5 square cm, what is the area of triangle DEF?

the ratio of areas of two similar shapes is proportional to the square of their corresponding sides

so area DEF/5 = 6^2 / 2^2 = 36/4 = 9/1
area DEF = 45

To find the area of triangle DEF, we can use the fact that triangles DEF and ABC are similar.

When two triangles are similar, the ratios of their corresponding sides are equal. In this case, the ratio of the lengths of sides DE to AB is 6 cm to 2 cm, or 3:1.

Since the area of a triangle is proportional to the square of its side lengths, we can use the ratio of the side lengths to find the ratio of the areas.

Let's calculate the ratio of the areas of the two triangles:

Ratio of areas = (Ratio of side lengths)^2 = (DE/AB)^2 = (6 cm/2 cm)^2 = (3)^2 = 9.

Therefore, the area of triangle DEF is 9 times the area of triangle ABC.

Since the area of triangle ABC is given as 5 square cm, we can find the area of triangle DEF as follows:

Area of triangle DEF = Area of triangle ABC × Ratio of areas = 5 square cm × 9 = 45 square cm.

So, the area of triangle DEF is 45 square cm.

To find the area of triangle DEF, we can use the concept of similarity between triangles. If two triangles are similar, the ratio of their corresponding sides is equal.

In this case, we know that triangle DEF is similar to triangle ABC. This means that the ratio of their corresponding sides is the same. Let's assume this ratio is x.

From the given information, we know that side DE is 6 cm and side AB is 2 cm. The ratio of the sides DE to AB is 6/2 = 3/1 = 3.

So, the ratio of the sides in the similar triangles DEF to ABC is 3. This means that the length of side EF is 3 times the length of side BC. If side BC has a length of y cm, then side EF has a length of 3y cm.

Now, let's find the area of triangle DEF. The area of a triangle can be found using the formula A = (1/2) * base * height.

In triangle DEF, the base is side EF, which has a length of 3y cm. The height can be any perpendicular segment from side EF to side DE.

Since triangle DEF is similar to triangle ABC, we can say that the height in triangle DEF is x times the height in triangle ABC.

Let's assume the height in triangle ABC is h cm. Then, the height in triangle DEF is xh cm.

Now, we can write the equation for the areas of the two triangles:

Area of triangle DEF = (1/2) * base * height = (1/2) * 3y * xh = (3/2) * xyh

We are given that the area of triangle ABC is 5 square cm. So, we can set up the equation:

5 = (1/2) * 2 * h

Simplifying, we get:

5 = h

Now that we know the height of triangle ABC is 5 cm, we can substitute this value back into the equation for the area of triangle DEF:

Area of triangle DEF = (3/2) * xy * 5

To find the area of triangle DEF, we need to know the value of x and y. Unfortunately, the given information does not provide this information, so we cannot find the exact area of triangle DEF. We can only express it in terms of x and y.