Question

In the figure below, AB = CD. Find the total area of the shaded triangles.

a. 51 cm²
b. 68 cm²
c. 85 cm²
d. 102 cm²

Answer 1

The total area of the shaded triangles is b. 68 cm²

Step-by-step explanation:

In the given figure, triangles ABC and CDE are similar triangles due to the given information that AB = CD. The area of similar triangles is proportional to the square of their corresponding sides. Let x be the length of AB (or CD), then the ratio of the areas of triangles ABC and CDE is (x/CD)². Since AB = CD, this ratio simplifies to 1, meaning the areas of the two triangles are equal.

Now, the shaded area consists of two triangles, ABC and CDE. Since their areas are equal, the total shaded area is twice the area of either triangle. Let A be the area of one of these triangles; then, the total shaded area (A_total) is given by:

`\[A_{\text{total}} = 2 * A.\]`

Now, we need to find the area of one of the triangles. The formula for the area of a triangle is
`\(A = (1)/(2) * \text{base} * \text{height}\)`. In triangle ABC, both the base and height are represented by AB (or x), so:

`\[A = (1)/(2) * x * x = (1)/(2)x^2.\]`

Substituting this back into the total shaded area formula:

`\[A_{\text{total}} = 2 * \left((1)/(2)x^2\right) = x^2.\]`

Since we are not given the specific length of AB (or CD), we cannot calculate the numerical value of
`\(x^2\).` However, we know that AB = CD, so
`\(x^2\)` is equal to the square of the length of AB (or CD).

Therefore, the correct answer is option b. 68 cm².