Proving Shapes in Coordinate Plane

The diagram shows quadrilateral ABCD. Which of the following would prove that ABCD is a parallelogram?

• Slope of 𝐴𝐷 = Slope of 𝐵𝐶

  Length of 𝐴𝐷 = Length of 𝐵𝐶

• Slope of 𝐴𝐷 = Slope of 𝐵𝐶

  Length of 𝐴B = Length of AD

• Length of 𝐴𝐷 = Length of 𝐵𝐶 = Length of DC

• Length of 𝐴𝐷 = Length of 𝐵𝐶 = Length of AB

Given the coordinates of A(3, 6), B(5, 2), and C(9, 4), which coordinates for D make ABCD a square?

• A. (6, 7)

• B. (7, 8)

• C. (7, 9)

• D. (8, 7)

Jillian and Tammy are considering a quadrilateral 𝐴𝐵𝐶𝐷. Their task is to prove it is a square.

Jillian says, “We just need to show that the slope of 𝐴𝐵 equals the slope of 𝐶𝐷 and the slope of 𝐵𝐶 equals the slope 𝐴𝐷.”

Tammy says, “We should show that 𝐴𝐶=𝐵𝐷 and that (𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝐴𝐶)×(𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝐵𝐷)=−1."

Whose method of proof is valid?

• A. Only Jillian’s is valid.

• B. Only Tammy’s is valid.

• C. Both are valid.

• D. Neither is valid.

Which type of triangle has vertices at the points 𝑅(2,1), 𝑆 (2,5), and 𝑇(4,1)?

• A. right

• B. acute

• C. isosceles

• D. equilateral

The vertices of a quadrilateral are 𝑀(–1,1),𝑁(1,–2),𝑂(5,0), and 𝑃(3,3). Which statement describes Quadrilateral MNOP?

• A. Quadrilateral MNOP is a rectangle.

• B. Quadrilateral MNOP is a trapezoid.

• C. Quadrilateral MNOP is a rhombus but not a square.

• D. Quadrilateral MNOP is a parallelogram but not a rectangle.

Triangle ABC has vertices with 𝐴(𝑥,3), 𝐵(−3,−1), and 𝐶(−1,−4). Determine and state a value of 𝑥 that would make triangle a right triangle.

In square 𝐺𝐸𝑂𝑀, the coordinates of 𝐺 are (2,−2) and the coordinates of 𝑂 are (−4,2). Determine and state the coordinates of vertices 𝐸 and 𝑀.

• (3, -1)

• (1, 3)

• (-3, -3)

• (3, 1)

• (-1, 3)

On a coordinate plane, a shape is plotted with vertices of (3, 1), (0, 4), (3, 7), and (6, 4). What is the area of the shape if each grid unit equals one centimeter?

Find the perimeter of the triangle to the nearest whole unit.

A triangle is shown on the coordinate plane below. What is the area of the triangle?

• A. 12 square units

• B. 24 square units

• C. 36 square units

• D. 48 square units