Question

problem solving with the coordinate plane
the figure factory

1. draw a rectangle in the coordinate plane that satisfies each of the following criteria. identify the ordered pair of each of the rectangles vertices.

• each of the vertices lies in a different quadrant.
• the perimeter of the rectangle is 38 units.
• either the x - axis or the y - axis, but not both, is a line of symmetry.

coordinate plane grid is shown with x - axis from - 10 to 10 and y - axis from - 10 to 10

Explanation:

Step1: Set perimeter formula

The perimeter of a rectangle is $P=2(l+w)$. Given $P=38$, so:
$$2(l+w)=38$$
$$l+w=19$$
We choose $l=10$, $w=9$ (any pair summing to 19 works).

Step2: Apply x-axis symmetry

For x-axis symmetry, if a vertex is $(a,b)$, its symmetric point is $(a,-b)$. To place vertices in 4 quadrants, we use:

• Quadrant 1: $(10,9)$
• Quadrant 4: $(10,-9)$
• Quadrant 2: $(-5,9)$
• Quadrant 3: $(-5,-9)$

Check: Horizontal side length $=10-(-5)=15$, vertical side length $=9-(-9)=18$? No, correct pair: Let $l=12$, $w=7$ (since $12+7=19$).

• Quadrant 1: $(12,7)$
• Quadrant 4: $(12,-7)$
• Quadrant 2: $(-7,7)$
• Quadrant 3: $(-7,-7)$

Horizontal length $=12-(-7)=19$? No, fix: For x-axis symmetry, left vertices have same $x$-negative, right same $x$-positive; top same $y$-positive, bottom same $y$-negative.
Let horizontal side $=a$, vertical side $=b$, $2(a+b)=38 \implies a+b=19$. Let $a=10$, $b=9$.

• Quadrant 1: $(5,9)$
• Quadrant 4: $(5,-9)$
• Quadrant 2: $(-5,9)$
• Quadrant 3: $(-5,-9)$ → No, this has both axes symmetry. Instead, use only x-axis symmetry, so left vertices have different $x$-values than right? No: Rectangle requires opposite sides equal. Correct: Use x-axis symmetry, so top two vertices in Q1/Q2, bottom two in Q4/Q3, with Q1 and Q4 share same $x$-positive, Q2 and Q3 share same $x$-negative; Q1 and Q2 share same $y$-positive, Q4 and Q3 share same $y$-negative.

Let $x_1=6$, $x_2=-3$, $y=5$. Then horizontal side length $=6-(-3)=9$, vertical side length $=5-(-5)=10$. Perimeter $=2(9+10)=38$, which fits.

Step3: Assign vertices to quadrants

• Quadrant 1: $(6,5)$
• Quadrant 2: $(-3,5)$
• Quadrant 3: $(-3,-5)$
• Quadrant 4: $(6,-5)$

Check: Each vertex in different quadrant, perimeter $2(9+10)=38$, x-axis is symmetry (reflect over x-axis maps vertices to each other), y-axis is not symmetry (reflect $(6,5)$ over y-axis is $(-6,5)$, not a vertex).

Answer:

One valid set of vertices is:

• Quadrant 1: $(6, 5)$
• Quadrant 2: $(-3, 5)$
• Quadrant 3: $(-3, -5)$
• Quadrant 4: $(6, -5)$

(When plotted, these form a rectangle meeting all criteria; other valid pairs exist as long as $l+w=19$, only one axis is symmetry, and vertices are in 4 distinct quadrants.)