Question

mixed review

1. a square with an area of 1 square meter is decomposed into 9 identical smaller squares. each smaller square is decomposed into two identical triangles.

a. what is the area, in square meters, of 6 triangles? if you get stuck, consider drawing a diagram.
b. how many triangles are needed to compose a region that is 1 1/2 square meters?

1. which shape has a larger area: a rectangle that is 7 inches by 3/4 inch, or a square with side - length of 2 1/2 inches? show your reasoning.

Explanation:

1.

Step1: Find area of one small square

The large - square has area $A = 1$ square meter and is decomposed into 9 identical smaller squares. So the area of one small square is $a_1=\frac{1}{9}$ square meters.

Step2: Find area of one triangle

Each small square is decomposed into 2 identical triangles. So the area of one triangle is $a_2=\frac{1}{9}\div2=\frac{1}{18}$ square meters.

Step3: Find area of 6 triangles

The area of 6 triangles is $A_1 = 6\times\frac{1}{18}=\frac{1}{3}$ square meters.

Step4: Find number of triangles for $1\frac{1}{2}$ square - meter region

First, convert $1\frac{1}{2}$ to an improper fraction: $1\frac{1}{2}=\frac{3}{2}$ square meters. Let the number of triangles be $n$. Since the area of one triangle is $\frac{1}{18}$ square meters, we have the equation $\frac{1}{18}n=\frac{3}{2}$. Solving for $n$ gives $n=\frac{3}{2}\div\frac{1}{18}=\frac{3}{2}\times18 = 27$.

Step1: Calculate area of the rectangle

The area of a rectangle with length $l = 7$ inches and width $w=\frac{3}{4}$ inch is $A_{r}=l\times w=7\times\frac{3}{4}=\frac{21}{4}=5.25$ square inches.

Step2: Calculate area of the square

The side - length of the square is $s = 2\frac{1}{2}=\frac{5}{2}$ inches. The area of the square is $A_{s}=s^{2}=(\frac{5}{2})^{2}=\frac{25}{4}=6.25$ square inches.

Step3: Compare the areas

Since $6.25>5.25$, the square has a larger area.

Answer:

a. $\frac{1}{3}$ square meters
b. 27

2.