Question

uncommon
mixed review

1. in the picture below, the area is 140 square millimeters. what is the length of the base?

diagram of parallelogram with height 28 mm

1. which shape has a larger area: a rectangle that is 7 inches by \\(\frac{3}{4}\\) inch, or a square with side length of \\(2\frac{1}{2}\\) inches? show your reasoning.

1. here is an octagon. (note: the diagonal sides of the octagon are not 4 inches long.)

diagram of octagon inside a square with dimensions 3 in, 4 in, 3 in, 3 in, 4 in, 3 in
find the exact area of the octagon. show your reasoning.

*adapted from illustrative math
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\the only way to learn mathematics is to do mathematics.\
16

Explanation:

Problem 1

Step1: Recall area formula for parallelogram

The area \( A \) of a parallelogram is given by \( A = base \times height \). Let the base be \( b \), height \( h = 28 \) mm, and area \( A = 140 \) square mm. So the formula is \( 140 = b \times 28 \).

Step2: Solve for base

To find \( b \), we divide both sides of the equation by 28: \( b=\frac{140}{28} \).

Step1: Calculate area of rectangle

The area \( A_{rect} \) of a rectangle is \( length \times width \). Here, length \( l = 7 \) inches and width \( w=\frac{2}{4}=\frac{1}{2} \) inch. So \( A_{rect}=7\times\frac{1}{2}=\frac{7}{2} = 3.5 \) square inches.

Step2: Calculate area of square

The area \( A_{square} \) of a square is \( side \times side \). The side length is \( 2\frac{1}{2}=\frac{5}{2} \) inches. So \( A_{square}=\frac{5}{2}\times\frac{5}{2}=\frac{25}{4} = 6.25 \) square inches.

Step3: Compare the two areas

Since \( 3.5<6.25 \), the square has a larger area.

Step1: Determine the dimensions of the outer square

First, find the side length of the outer square. The horizontal segments are \( 3 + 4+ 3=10 \) inches, and the vertical segments are \( 3 + 4 + 3 = 10 \) inches. So the outer square has side length \( s = 10 \) inches, and its area \( A_{square}=10\times10 = 100 \) square inches.

Step2: Calculate the area of the four right - angled triangles

Each of the four corner triangles is a right - angled triangle with legs of length \( 3 \) inches. The area of one right - angled triangle is \( A_{triangle}=\frac{1}{2}\times base\times height=\frac{1}{2}\times3\times3=\frac{9}{2} \) square inches. The area of four such triangles is \( 4\times\frac{9}{2}=18 \) square inches.

Step3: Calculate the area of the octagon

The area of the octagon \( A_{octagon} \) is the area of the outer square minus the area of the four triangles. So \( A_{octagon}=A_{square}-A_{triangles}=100 - 18=82 \) square inches.

Answer:

The length of the base is \( 5 \) millimeters.

Problem 2