Question

13 cm 5 cm 2 cm 5 cm 14 cm what is the area of the composite figure? 70 cm² 100 cm² 105 cm² 130 cm²

Explanation:

Answer:

To find the area of the composite figure, we can split it into a trapezoid and a rectangle (or use other methods, but here's one way):

1. Trapezoid part: The trapezoid has bases \( 5 \, \text{cm} \) and \( 5 + 5 = 10 \, \text{cm} \), and the height (horizontal length) is \( 14 - 2 = 12 \, \text{cm} \).

Area of trapezoid: \( \frac{1}{2} \times (5 + 10) \times 12 = 90 \, \text{cm}^2 \).

1. Rectangle part: The rectangle has length \( 2 \, \text{cm} \) and height \( 5 \, \text{cm} \).

Area of rectangle: \( 2 \times 5 = 10 \, \text{cm}^2 \).

1. Total area: \( 90 + 10 = 100 \, \text{cm}^2 \)? Wait, no—wait, maybe another split. Wait, actually, the correct split: the figure can be a trapezoid with bases \( 5 \, \text{cm} \) and \( 5 + 5 = 10 \, \text{cm} \), and the horizontal length is \( 14 - 2 = 12 \, \text{cm} \), plus a rectangle of \( 2 \times 5 \). Wait, no, maybe I made a mistake. Wait, let's check again.

Alternative method: The figure can be a trapezoid with bases \( 5 \, \text{cm} \) and \( 5 + 5 = 10 \, \text{cm} \), and the height (the horizontal distance) is \( 14 - 2 = 12 \, \text{cm} \), and then a rectangle of \( 2 \times 5 \). Wait, no, actually, the correct area is \( 105 \, \text{cm}^2 \)? Wait, no—wait, let's calculate the area of the trapezoid with bases \( 5 \) and \( 5 + 5 = 10 \), and the horizontal length is \( 14 - 2 = 12 \), so area of trapezoid is \( \frac{(5 + 10)}{2} \times 12 = 90 \). Then the rectangle is \( 2 \times 5 = 10 \), total \( 100 \)? But the options include \( 105 \). Wait, maybe another approach: the figure is a trapezoid with bases \( 5 \) and \( 5 + 5 = 10 \), and the height is \( 14 \), but no, the right side has a notch. Wait, let's use the formula for the area of a trapezoid plus a rectangle. Wait, the bottom base is \( 14 \), the left height is \( 5 \), the right part has a vertical segment of \( 5 \) and a horizontal segment of \( 2 \). So the upper base of the trapezoid (if we consider the main trapezoid) is \( 5 + 5 = 10 \)? No, wait, the slant side is \( 13 \). Let's check the horizontal distance between the two vertical sides (left \( 5 \) and right \( 5 + 5 = 10 \)): using Pythagoras, \( 13^2 - (10 - 5)^2 = 169 - 25 = 144 \), so horizontal distance is \( 12 \), which matches \( 14 - 2 = 12 \). So the trapezoid has bases \( 5 \) and \( 10 \), height \( 12 \), area \( 90 \). Then the rectangle is \( 2 \times 5 = 10 \), total \( 100 \)? But the options have \( 105 \). Wait, maybe I split wrong. Let's try another way: the figure is a trapezoid with bases \( 5 \) and \( 5 + 5 = 10 \), and the height is \( 14 \), but no, the notch is \( 2 \) cm. Wait, no—wait, the correct area is \( 105 \, \text{cm}^2 \)? Wait, let's calculate the area as a trapezoid with bases \( 5 \) and \( 5 + 5 = 10 \), and the height is \( 14 - 2 = 12 \), plus a rectangle of \( 2 \times (5 + 5) = 20 \)? No, that's not right. Wait, maybe the figure is a trapezoid with bases \( 5 \) and \( 5 + 5 = 10 \), and the height is \( 14 \), minus the area of the notch? No, the notch is a rectangle of \( 2 \times (10 - 5) = 10 \)? No, I'm confused. Wait, let's look at the options. The correct answer is \( 105 \, \text{cm}^2 \)? Wait, no—wait, let's do it step by step.

The composite figure can be divided into a trapezoid and a rectangle. Wait, the left part is a trapezoid with bases \( 5 \) cm and \( 5 + 5 = 10 \) cm, and the horizontal length (height of trapezoid) is \( 14 - 2 = 12 \) cm. Area of trapezoid: \( \frac{(5 + 10)}{2} \times 12 = 90 \) cm². Then the right part is a rectangle with length \( 2 \) cm and height \( 5 + 5 = 10 \) cm? No, that's not. Wait, the right vertical segment is \( 5 \) cm, and the horizontal segment is \( 2 \) cm. So the rectangle is \( 2 \times 5 = 10 \) cm². Then total area is \( 90 + 10 = 100 \) cm²? But the option is \( 100 \) or \( 105 \). Wait, maybe I made a mistake in the trapezoid bases. Wait, the left height is \( 5 \) cm, the right height (the vertical side) is \( 5 + 5 = 10 \) cm? No, the right side has a vertical segment of \( 5 \) cm (the lower part) and \( 5 \) cm (the upper part), so total height \( 10 \) cm. The horizontal length is \( 14 \) cm, but there's a notch of \( 2 \) cm. So the trapezoid has bases \( 5 \) and \( 10 \), and the height (horizontal distance) is \( 14 - 2 = 12 \) cm. Then the rectangle is \( 2 \times 5 = 10 \) cm². So total area \( 90 + 10 = 100 \) cm². So the answer is \( 100 \, \text{cm}^2 \).

Wait, but let's check with another method. The figure can be considered as a trapezoid with bases \( 5 \) and \( 5 + 5 = 10 \), and the height is \( 14 \), minus the area of the rectangle that's missing. The missing rectangle would be \( 2 \times (10 - 5) = 10 \) cm². Area of trapezoid with bases \( 5 \) and \( 10 \), height \( 14 \): \( \frac{(5 + 10)}{2} \times 14 = 105 \) cm². Then subtract the missing rectangle: \( 105 - 10 = 95 \)? No, that's not. Wait, I'm getting confused. Let's use coordinates. Let's place the bottom left corner at \( (0, 0) \). Then:

• Bottom left: \( (0, 0) \)
• Bottom right: \( (14, 0) \)
• The right vertical segment: from \( (14, 0) \) up to \( (14, 5) \), then left to \( (14 - 2, 5) = (12, 5) \), then up to \( (12, 5 + 5) = (12, 10) \), then left to \( (0, 10) \)? No, the left side is \( (0, 5) \) to \( (0, 0) \), and the top left is \( (0, 5) \) to \( (12, 10) \) (since the slant side is \( 13 \), and the vertical difference is \( 10 - 5 = 5 \), horizontal difference \( 12 \), so \( 5^2 + 12^2 = 25 + 144 = 169 = 13^2 \), which matches). So the coordinates are:

• \( (0, 0) \)
• \( (14, 0) \)
• \( (14, 5) \)
• \( (12, 5) \)
• \( (12, 10) \)
• \( (0, 5) \)
• Back to \( (0, 0) \)

Now, to find the area, we can use the shoelace formula. Let's list the coordinates in order:

\( (0, 0) \), \( (14, 0) \), \( (14, 5) \), \( (12, 5) \), \( (12, 10) \), \( (0, 5) \), \( (0, 0) \)

Shoelace formula:

Sum of \( x_i y_{i+1} \):

\( 0 \times 0 + 14 \times 5 + 14 \times 5 + 12 \times 10 + 12 \times 5 + 0 \times 0 \)
= \( 0 + 70 + 70 + 120 + 60 + 0 = 320 \)

Sum of \( y_i x_{i+1} \):

\( 0 \times 14 + 0 \times 14 + 5 \times 12 + 5 \times 12 + 10 \times 0 + 5 \times 0 \)
= \( 0 + 0 + 60 + 60 + 0 + 0 = 120 \)

Area = \( \frac{1}{2} |320 - 120| = \frac{1}{2} \times 200 = 100 \, \text{cm}^2 \).

So the correct answer is \( 100 \, \text{cm}^2 \), which is the second option.