Question

how can you find the total area of this shape in square centimeters? select all the correct answers. a add 4 × 2 to 3 × 7. b add 7 × 2 to 3 × 7.

Explanation:

Step1: Analyze Option A

The shape can be divided into two rectangles. One rectangle could have dimensions \(4 \, \text{cm}\) (length) and \(2 \, \text{cm}\) (width), and the other could have dimensions \(3 \, \text{cm}\) (width) and \(7 \, \text{cm}\) (height). The area of the first rectangle is \(4\times2\) and the second is \(3\times7\). Adding them gives the total area. But wait, actually, let's re - check the dimensions. Wait, the top rectangle: the length should be \(7 \, \text{cm}\) (since the top side is \(7 \, \text{cm}\)) and width \(2 \, \text{cm}\), and the bottom rectangle: width \(3 \, \text{cm}\) and height \(7 \, \text{cm}\). Wait, no, maybe another way. Wait, the vertical rectangle: width \(3 \, \text{cm}\), height \(7 \, \text{cm}\), area \(3\times7\). The horizontal rectangle: length \(7 \, \text{cm}\) (since the top is \(7 \, \text{cm}\)) and width \(2 \, \text{cm}\), area \(7\times2\). Wait, but option A says \(4\times2\). Wait, maybe I made a mistake. Wait, the horizontal part: the length of the horizontal rectangle (the top one) is \(7 \, \text{cm}\), but if we look at the indentation, the length of the horizontal rectangle (the top part) can also be considered as \(4 + 3=7 \, \text{cm}\), and width \(2 \, \text{cm}\). The vertical rectangle: width \(3 \, \text{cm}\), height \(7 \, \text{cm}\). So the area of the top rectangle is \(7\times2\) and the vertical one is \(3\times7\). But option A is \(4\times2+3\times7\). Wait, \(4 + 3=7\), so \(4\times2+3\times2=(4 + 3)\times2 = 7\times2\), no, \(4\times2\) is not equal to \(7\times2\). Wait, maybe the correct way is: the shape can be divided into a top rectangle (length \(7 \, \text{cm}\), width \(2 \, \text{cm}\)) and a bottom rectangle (width \(3 \, \text{cm}\), height \(7 \, \text{cm}\)). So area of top is \(7\times2\), area of bottom is \(3\times7\). So option B: \(7\times2+3\times7\) is correct. Wait, but let's check option A again. If we take the top rectangle as length \(4 \, \text{cm}\) and width \(2 \, \text{cm}\), and the right - hand rectangle as width \(3 \, \text{cm}\) and height \(7 \, \text{cm}\), but then we are missing the part between \(4 \, \text{cm}\) and \(7 \, \text{cm}\) (which is \(3 \, \text{cm}\)). So actually, the correct division is top rectangle (length \(7 \, \text{cm}\), width \(2 \, \text{cm}\)) and bottom rectangle (width \(3 \, \text{cm}\), height \(7 \, \text{cm}\)). So the area of top is \(7\times2\), area of bottom is \(3\times7\), so option B is correct. Wait, but maybe the question has a typo or my analysis is wrong. Wait, let's calculate the area in two ways.

First way: Divide the shape into a top rectangle (length \(7 \, \text{cm}\), width \(2 \, \text{cm}\)) and a bottom rectangle (width \(3 \, \text{cm}\), height \(7 \, \text{cm}\)). Area of top: \(7\times2 = 14\), area of bottom: \(3\times7=21\), total area \(14 + 21=35\).

Second way: Divide into a left - indent rectangle (length \(4 \, \text{cm}\), width \(2 \, \text{cm}\)) and a right - hand rectangle (width \(3 \, \text{cm}\), height \(7 \, \text{cm}\)) and a middle part? No, that's not right. Wait, \(4\times2+3\times7 = 8+21 = 29\), which is wrong. \(7\times2+3\times7=14 + 21 = 35\). So option B is correct. Wait, but the original problem says "select all the correct answers". Maybe I made a mistake in the division. Wait, another way: the shape is a \(7\times7\) square with a \(4\times5\) rectangle cut out? No, \(7\times7=49\), \(4\times5 = 20\), \(49-20 = 29\), which is not matching. Wait, no, the height of the vertical rectangle is \(7 \, \text{cm}\), the width is \(3 \, \text{cm}\), area \(21\). The horizontal rectangle: length \(7 \, \text{cm}\), width \(2 \, \text{cm}\), area \(14\), total \(35\). If we do \(4\times2+3\times7=8 + 21=29\), which is wrong. So option B is correct. Wait, but maybe the first division is wrong. Wait, the vertical rectangle: width \(3 \, \text{cm}\), height \(7 \, \text{cm}\), area \(21\). The horizontal rectangle: length \(4 \, \text{cm}\), width \(2 \, \text{cm}\), area \(8\), and then there is another rectangle? No, the shape is a single connected shape. Wait, I think I messed up the dimensions. Let's look at the figure again:

The top horizontal side is \(7 \, \text{cm}\), the left side of the top part is \(2 \, \text{cm}\) (height), then there is an indentation of \(4 \, \text{cm}\) (length) and \(5 \, \text{cm}\) (height), and the right side has a width of \(3 \, \text{cm}\) and height of \(7 \, \text{cm}\).

So the area can be calculated as the area of the big rectangle (if we consider the outer dimensions) minus the area of the indentation. The outer rectangle would be \(7 \, \text{cm}\) (width) and \(7 \, \text{cm}\) (height), area \(7\times7 = 49\). The indentation is a rectangle with length \(4 \, \text{cm}\) and height \(5 \, \text{cm}\), area \(4\times5 = 20\). So total area \(49-20 = 29\). Wait, now this is different. So where is the mistake?

Wait, the height of the indentation: the total height is \(7 \, \text{cm}\), the height of the top part is \(2 \, \text{cm}\), so the height of the indentation is \(7 - 2=5 \, \text{cm}\), and the length of the indentation is \(4 \, \text{cm}\). So the area of the indentation is \(4\times5 = 20\). The area of the outer rectangle (if we consider the shape as a \(7\times7\) square with a \(4\times5\) cut - out) is \(7\times7-4\times5=49 - 20 = 29\).

Now let's check the options:

Option A: \(4\times2+3\times7=8 + 21 = 29\)

Option B: \(7\times2+3\times7=14 + 21 = 35\)

Ah! Now I see. So the correct way is to divide the shape into two rectangles: one is the vertical rectangle with width \(3 \, \text{cm}\) and height \(7 \, \text{cm}\) (area \(3\times7 = 21\)), and the other is the horizontal rectangle with length \(4 \, \text{cm}\) and width \(2 \, \text{cm}\) (area \(4\times2 = 8\)), and also, wait, no, \(4\times2+3\times7=29\), which matches the area when we calculate it as \(7\times7 - 4\times5=29\). And \(7\times2+3\times7 = 35\), which is wrong. So option A is correct.

Wait, let's re - verify:

Vertical rectangle: width \(3 \, \text{cm}\), height \(7 \, \text{cm}\), area \(3\times7 = 21\)

Horizontal rectangle: length \(4 \, \text{cm}\), width \(2 \, \text{cm}\), area \(4\times2 = 8\)

Total area \(21 + 8=29\)

Which is equal to \(7\times7-4\times5=49 - 20 = 29\)

And option B: \(7\times2+3\times7=14 + 21 = 35 eq29\)

So option A is correct.

Step2: Analyze Option B

As calculated above, \(7\times2+3\times7 = 14 + 21 = 35\), which is not equal to the correct total area of \(29\) (when calculated as \(7\times7-4\times5\) or \(4\times2+3\times7\)). So option B is incorrect.

Answer:

A. Add \(4\times2\) to \(3\times7\)