Question

In similar triangles ABC and CDE below, CD=DE=8 . If BC=4 , what is the area of ABC?

Answer 1

The area of the larger square is four times that of the smaller one, as the area ratio of similar figures is the square of the scale factor, which is 2² or 4 in this example.

Step-by-step explanation:

The question asks about the area of similar triangles and how the area of a larger square compares to that of a smaller square when its dimensions are twice as large.

To solve this problem, we can use the fact that the ratio of the areas of similar figures is the square of the scale factor. In the case of the squares mentioned, if the side of the first square is 4 inches and the larger one has sides twice as long, the side of the larger square would be 4 inches × 2, which equals 8 inches.

The area of the first (smaller) square is calculated by squaring its side length: Area_small = side × side = 4 in × 4 in = 16 in².

The area of the second (larger) square is calculated similarly: Area_large = 8 in × 8 in = 64 in².

Since 64 in² is exactly four times larger than 16 in², we can conclude that the area of the larger square is four times the area of the smaller square. This illustrates the rule that when comparing areas of similar figures, the ratio of their areas is the square of the scale factor (2² = 4 in this case).

Answer 2

In similar triangles ABC and CDE, CD=DE=8, if BC=4, the area of triangle ABC is 8 square units.

Step-by-step explanation:

In similar triangles, corresponding sides are proportional. Therefore, if CD=DE=8 and BC=4, we can set up the following proportion:

CD/BC = DE/AC

8/4 = 8/AC

Simplifying the proportion, we get:

2 = 8/AC

Multiplying both sides by AC:

2AC = 8

Dividing both sides by 2:

AC = 4

Now, to find the area of triangle ABC, we can use the formula for the area of a triangle:

Area = 1/2 * base * height

In triangle ABC, the base is BC = 4 and the height is AC = 4.

Plugging in these values, we get:

Area = 1/2 * 4 * 4 = 8 square units

So therefore the area of triangle ABC is 8 square units.