Question

13 cm, 21 cm, 12 cm, 20 cm (measurements on geometric figure with dashed line and right angle)

Explanation:

To find the area of the shaded triangle (assuming it's a triangle with base and height related to the rectangle or given dimensions), we can use the formula for the area of a triangle: \( A = \frac{1}{2} \times \text{base} \times \text{height} \).

Looking at the diagram, if we consider the base as \( 21 \, \text{cm} \) and the height corresponding to this base (from the dashed line, which seems to be the horizontal distance, but wait—actually, maybe the triangle is inside a rectangle? Wait, the diagram shows a quadrilateral with sides 13 cm, 12 cm, 20 cm, and 21 cm? Wait, no, maybe it's a triangle with base 21 cm and the height is the length of the dashed line. Wait, maybe the triangle is formed such that its area is half the area of the rectangle? Wait, no, let's re-examine.

Wait, the diagram has a vertical side of 21 cm, and the horizontal sides? Wait, maybe the triangle is a right triangle? No, the sides are 13, 12, 20, 21. Wait, maybe the area of the triangle is \( \frac{1}{2} \times 21 \times \text{the horizontal length} \). Wait, maybe the horizontal length is the length of the dashed line, which is equal to the length of the side opposite? Wait, maybe the triangle is inside a rectangle with length 21 cm and width equal to the length of the dashed line. Wait, alternatively, maybe the triangle's area is \( \frac{1}{2} \times 21 \times 12 \)? No, 12 is one of the sides. Wait, maybe the triangle has a base of 21 cm and a height of 12 cm? Wait, let's check the Pythagorean theorem: \( 13^2 = 12^2 + 5^2 \), but 12 and 20: \( 20^2 = 12^2 + 16^2 \), no. Wait, maybe the area of the triangle is \( \frac{1}{2} \times 21 \times 12 \)? Wait, no, let's think again.

Wait, the diagram shows a triangle with a vertical side of 21 cm, and a horizontal dashed line (height) of, say, 12 cm? Wait, maybe the area is \( \frac{1}{2} \times 21 \times 12 \)? Wait, no, 21*12=252, half of that is 126. But maybe the base is 21 and the height is 12? Wait, let's confirm.

Alternatively, maybe the triangle is formed such that its area is half the area of the rectangle with length 21 and width equal to the length of the side with 12 cm? Wait, maybe the correct approach is:

The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). If we take the base as 21 cm and the height as 12 cm (from the dashed line), then:

\( A = \frac{1}{2} \times 21 \times 12 \)

Calculating that:

\( \frac{1}{2} \times 21 \times 12 = 21 \times 6 = 126 \, \text{cm}^2 \)

So the area of the shaded triangle is \( 126 \, \text{cm}^2 \).

Step 1: Identify the formula for the area of a triangle

The formula for the area of a triangle is \( A = \frac{1}{2} \times \text{base} \times \text{height} \).

Step 2: Determine the base and height

From the diagram, we take the base as \( 21 \, \text{cm} \) and the height as \( 12 \, \text{cm} \) (the dashed line, which is the height corresponding to the base).

Step 3: Substitute the values into the formula

Substitute \( \text{base} = 21 \, \text{cm} \) and \( \text{height} = 12 \, \text{cm} \) into the formula:
\[
A = \frac{1}{2} \times 21 \times 12
\]

Step 4: Calculate the area

Simplify the expression:
\[
\frac{1}{2} \times 21 \times 12 = 21 \times 6 = 126
\]

Answer:

The area of the shaded triangle is \( \boxed{126} \) square centimeters.