Question

a parallelogram is cut out of a 12 - inch by 8 - inch sheet of paper. there are four right triangle remnants. two have the dimensions 2 inches by 9 inches, and the other two have the dimensions 3 inches by 6 inches. the resulting parallelogram has a base of approximately 9.22 inches. complete the following steps to calculate the altitude of the parallelogram using area methods. the area of the sheet of paper is square inches. the combined area of the triangle cutouts is square inches. the area of the parallelogram is square inches. the altitude of the parallelogram rounded to two decimals is square inches.

Explanation:

Step1: Calculate the area of the paper

The paper is a rectangle. The area formula for a rectangle is \(A = l\times w\). Here, \(l = 12\) inches and \(w=8\) inches.
\(A_{paper}=12\times8 = 96\) square inches.

Step2: Calculate the combined area of the triangle cut - outs

The area formula for a triangle is \(A=\frac{1}{2}bh\).
For the two triangles with \(b = 2\) inches and \(h = 9\) inches: \(A_1=2\times(\frac{1}{2}\times2\times9)= 18\) square inches.
For the two triangles with \(b = 3\) inches and \(h = 6\) inches: \(A_2=2\times(\frac{1}{2}\times3\times6)=18\) square inches.
The combined area of the triangle cut - outs \(A_{triangles}=A_1 + A_2=18 + 18=36\) square inches.

Step3: Calculate the area of the parallelogram

\(A_{parallelogram}=A_{paper}-A_{triangles}\)
\(A_{parallelogram}=96 - 36=60\) square inches.

Step4: Calculate the altitude of the parallelogram

The area formula for a parallelogram is \(A = bh\), where \(b = 9.22\) inches (base) and \(h\) is the altitude.
We know \(A = 60\) square inches. Rearranging the formula for \(h\), we get \(h=\frac{A}{b}\)
\(h=\frac{60}{9.22}\approx6.51\) inches.

Answer:

96, 36, 60, 6.51