In the standard (x, y) coordinate plane, a trapezoid has vertices at A(-5, -4), B(7, -4), … — ACT

In the standard $(x, y)$ coordinate plane, a trapezoid has vertices at $A(-5, -4)$ , $B(7, -4)$ , $C(7, 1)$ , and $D(-5, 6)$ . What is the perimeter of this trapezoid?

Answer: 40

Answer

The perimeter of the trapezoid is 40.

The perimeter of the trapezoid is found by summing the lengths of its four sides. Two sides are vertical (AD of length 10 and BC of length 5), one side is horizontal (AB of length 12), and the final side is skewed (CD of length 13, found using the distance formula). Summing these gives 10 + 5 + 12 + 13 = 40.

Step-by-Step Solution

Calculate the lengths of the vertical sides AD and BC

AD = 10, BC = 5

Since vertices A(-5, -4) and D(-5, 6) share the same x-coordinate of -5, the side AD is vertical. Its length is the difference in their y-coordinates: 6 - (-4) = 10. Similarly, vertices B(7, -4) and C(7, 1) share the same x-coordinate of 7, so the side BC is vertical. Its length is the difference in their y-coordinates: 1 - (-4) = 5.

Calculate the length of the horizontal side AB

AB = 12

Since vertices A(-5, -4) and B(7, -4) share the same y-coordinate of -4, the side AB is horizontal. Its length is the difference in their x-coordinates: 7 - (-5) = 12.

Calculate the length of the skewed side CD using the distance formula

CD = 13

Using the coordinates of C(7, 1) and D(-5, 6), the distance is calculated as: CD = \sqrt{(-5 - 7)^2 + (6 - 1)^2} = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.

Sum the lengths of all sides to find the perimeter

Perimeter = 40

The perimeter of a polygon is the sum of the lengths of all its sides: Perimeter = AB + BC + CD + DA = 12 + 5 + 13 + 10 = 40.

Key Concept

Calculating side lengths of a geometric figure on a coordinate plane to find its perimeter

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