Question

turner, daniya the area of trapezoid abcd is 96 square units. the coordinates of the trapezoid are given, where g and h are real numbers. • a(-7, g) • b(-2, h) • c(5, h) • d(10, g) what is the perimeter, in units, of trapezoid abcd? round your answer to the nearest tenth as needed.

Explanation:

Step1: Identify the bases and height of the trapezoid

First, we note that points \( B(-2, h) \) and \( C(5, h) \) have the same \( y \)-coordinate, so \( BC \) is a horizontal line segment. The length of \( BC \) is \( 5 - (-2)=7 \). Similarly, points \( A(-7, g) \) and \( D(10, g) \) have the same \( y \)-coordinate, so \( AD \) is a horizontal line segment. The length of \( AD \) is \( 10 - (-7)=17 \). The height of the trapezoid (the vertical distance between the two horizontal bases \( AD \) and \( BC \)) is \( |h - g| \).

The formula for the area of a trapezoid is \( A=\frac{(b_1 + b_2)}{2}\times h \), where \( b_1 \) and \( b_2 \) are the lengths of the two bases and \( h \) is the height. We know the area \( A = 96 \), \( b_1=17 \), \( b_2 = 7 \).

Step2: Calculate the height \( |h - g| \)

Substitute the values into the area formula:
\[
96=\frac{(17 + 7)}{2}\times|h - g|
\]
First, calculate \( \frac{17 + 7}{2}=\frac{24}{2} = 12 \)
So, \( 96=12\times|h - g| \)
Divide both sides by 12: \( |h - g|=\frac{96}{12}=8 \)

Step3: Calculate the lengths of the non - parallel sides \( AB \) and \( CD \)

For side \( AB \): The coordinates of \( A(-7,g) \) and \( B(-2,h) \). Using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \), we have \( x_2 - x_1=-2-(-7)=5 \) and \( y_2 - y_1=h - g \) (and \( |h - g| = 8 \), so \( (h - g)^2=64 \)). Then the length of \( AB \) is \( \sqrt{5^2+8^2}=\sqrt{25 + 64}=\sqrt{89}\approx9.43 \)

For side \( CD \): The coordinates of \( C(5,h) \) and \( D(10,g) \). Using the distance formula, \( x_2 - x_1=10 - 5 = 5 \) and \( y_2 - y_1=g - h \) (and \( (g - h)^2=(h - g)^2 = 64 \)). Then the length of \( CD \) is \( \sqrt{5^2+8^2}=\sqrt{25+64}=\sqrt{89}\approx9.43 \)

Step4: Calculate the perimeter

The perimeter \( P \) of the trapezoid is the sum of the lengths of all four sides: \( P=AD + BC+AB + CD \)
We know \( AD = 17 \), \( BC = 7 \), \( AB\approx9.43 \), \( CD\approx9.43 \)
So, \( P=17 + 7+9.43+9.43=17 + 7+18.86=42.86\approx42.9 \) (rounded to the nearest tenth)

Answer:

\( 42.9 \)