Question

lesson 2 practice problems

1. the diagonal of a rectangle is shown.

image of a grid with a blue rectangle and its diagonal
a. decompose the rectangle along the diagonal, and recompose the two pieces to make a different shape.
b. how does the area of this new shape compare to the area of the original rectangle? explain how you know.

1. priya decomposed a square into 16 smaller, equal - size squares and then cut out 4 of the small squares and attached them around the outside of original square to make a new figure.

how does the area of her new figure compare with that of the original square?
images of original square and new figure
a. the area of the new figure is greater.
b. the two figures have the same area.
c. the area of the original square is greater.
d. we dont know because neither the side length nor the area of the original square is known.

Explanation:

Question 1a

Step1: Identify the two triangles

The rectangle is split into two congruent right - angled triangles by the diagonal.

Step2: Recompose the triangles

We can place the two triangles next to each other such that their hypotenuses (the diagonal of the rectangle) are adjacent. For example, if the rectangle has length \( l \) and width \( w \), and the diagonal divides it into triangle 1 with vertices \((0,0)\), \((l,0)\), \((0,w)\) and triangle 2 with vertices \((l,0)\), \((0,w)\), \((l,w)\), we can move triangle 1 so that its side of length \( l \) is adjacent to the side of length \( l \) of triangle 2 but in a different orientation. This will form a parallelogram (or other shapes depending on the orientation).

Question 1b

Step1: Recall the concept of area conservation

When we decompose a figure into parts and then recompose those parts to form a new figure, the total area of the parts remains the same.

Step2: Apply to the rectangle and new shape

The original rectangle is decomposed into two triangles. When we recompose these two triangles to form a new shape, the area of the new shape is equal to the sum of the areas of the two triangles. Since the sum of the areas of the two triangles is equal to the area of the original rectangle (because the area of a rectangle \( A = lw\) and the area of each triangle is \(\frac{1}{2}lw\), so two triangles have an area of \( 2\times\frac{1}{2}lw=lw\)), the area of the new shape is equal to the area of the original rectangle.

Question 2

Step1: Analyze the area of the original square

Let the area of each small square be \( A_s \). The original square is composed of 16 small squares, so the area of the original square \( A_{original}=16A_s \).

Step2: Analyze the area of the new figure

Priya cuts out 4 small squares from the original square and attaches them around the outside. The number of small squares in the new figure is still \( 16 - 4+4 = 16\) (we subtract the 4 we cut out and then add the 4 we attached). So the area of the new figure \( A_{new}=16A_s \).

Step3: Compare the areas

Since \( A_{original}=16A_s \) and \( A_{new}=16A_s \), the two figures have the same area.

Answer:

s:
1a. (Description of the new shape: For example, a parallelogram formed by placing the two triangular parts with their hypotenuses together. The actual drawing would show the rectangle split into two triangles and then the triangles rearranged, e.g., if the rectangle is 3 units long and 2 units wide, the two triangles (each with base 3 and height 2) can be arranged to form a parallelogram with base 3 and height 2 or other equivalent shapes).
1b. The area of the new shape is equal to the area of the original rectangle. This is because the new shape is made by rearranging the two triangular parts of the original rectangle, and the total area of the parts (the two triangles) remains the same as the area of the rectangle.

1. B. The two figures have the same area.