A parallelogram with vertices P(-3, -2), Q(5, -2), R(8, 2), and S(0, 2) is plotted on a st… — ACT

A parallelogram with vertices $P(-3, -2)$ , $Q(5, -2)$ , $R(8, 2)$ , and $S(0, 2)$ is plotted on a standard coordinate grid. What is the perimeter of parallelogram $PQRS$ ?

A
$16 + 2\sqrt{7}$
26Answer
C
30
D
32
E
34

Answer

The correct answer is 26, which is the perimeter of the parallelogram.

The horizontal base segments have a length of

5 - (-3) = 8

. The slanted side has a length of

\sqrt{(0 - (-3))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = 5

. Summing the four sides gives a perimeter of

2(8) + 2(5) = 26

Step-by-Step Solution

Calculate the length of the horizontal base PQ using the x-coordinates of P(-3, -2) and Q(5, -2).

PQ = 5 - (-3) = 8

Horizontal segments on the coordinate plane have a length equal to the difference in their x-coordinates.

Apply the distance formula to find the length of the slanted side PS with vertices P(-3, -2) and S(0, 2).

PS = \sqrt{(0 - (-3))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5

The distance formula determines the straight-line distance between two coordinates.

Calculate the total perimeter of parallelogram PQRS by summing the lengths of all four sides.

\text{Perimeter} = 2 \times \text{base} + 2 \times \text{slanted side} = 2(8) + 2(5) = 16 + 10 = 26

The perimeter of a parallelogram is the sum of all its outer boundary lengths.

Key Concept

Calculating the perimeter of geometric figures on the coordinate plane by determining horizontal distances and applying the distance formula for skewed lines.

Alternative Method

Instead of using the distance formula, you can sketch a right triangle for the slanted side PS. The horizontal leg has length 3 and the vertical leg has length 4. Recognizing this as a standard 3-4-5 right triangle, the hypotenuse (the slanted side) must be 5.

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic