Exercises: Use coordinates to prove theorems - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

These problems review formulas you will need for coordinate geometry proofs.

1.

What is the midpoint of the segment with endpoints $(0, 0)$ and $(a, b)$ ?

A.

$(a, b)$

B.

$\left(\dfrac{a}{2}, \dfrac{b}{2}\right)$

C.

$\left(\dfrac{a}{2}, b\right)$

D.

$(2a, 2b)$

2.

Find the distance between the points $(1, 2)$ and $(4, 6)$ . Enter the exact value.

3.

A segment has slope $\dfrac{3}{4}$ . What is the slope of a line perpendicular to it?

A.

$\dfrac{3}{4}$

B.

$-\dfrac{3}{4}$

C.

$-\dfrac{4}{3}$

D.

$\dfrac{4}{3}$

B

Fluency Practice

Use the distance and slope formulas to answer each question. Show your computations.

1.

The four vertices of a quadrilateral are $A(0, 0)$ , $B(4, 0)$ , $C(4, 3)$ , and $D(0, 3)$ .
Find the length of side $AB$ .

2.

Using the same quadrilateral $A(0, 0)$ , $B(4, 0)$ , $C(4, 3)$ , $D(0, 3)$ from the previous problem, find the slope of side $BC$ .

3.

Consider the four points $P(-3, -1)$ , $Q(1, -1)$ , $R(3, 3)$ , $S(-1, 3)$ .
How many pairs of opposite sides have equal slopes? Enter a whole number.

4.

For the quadrilateral with vertices $A(2, 1)$ , $B(6, 3)$ , $C(5, 5)$ , $D(1, 3)$ , compute $AB^2$ (the square of the length of side $AB$ ). Enter a whole number.

5.

The four vertices of a quadrilateral are $P(0, 0)$ , $Q(4, 1)$ , $R(5, 5)$ , $S(1, 4)$ .
Compute the product of the slopes of sides $PQ$ and $QR$ . What does this product tell you?
Enter the numerical value of the product only.

C

Mixed Practice

These problems test coordinate proof skills in different ways.

D

Coordinate Proof Problems

Show all computations and state your conclusion with justification.

1.

A rectangle has vertices at $A(0, 0)$ , $B(a, 0)$ , $C(a, b)$ , and $D(0, b)$ where $a > 0$ and $b > 0$ .

1.

Find the midpoint of diagonal $AC$ . Enter the $x$ -coordinate only (in terms of $a$ and $b$ ).

2.

The midpoint of diagonal $BD$ (from $B(a,0)$ to $D(0,b)$ ) is $\left(\dfrac{a}{2}, \dfrac{b}{2}\right)$ .
What does this prove about the diagonals of the rectangle?

A.

The diagonals are congruent.

B.

The diagonals are perpendicular.

C.

The diagonals bisect each other.

D.

The diagonals are parallel.

2.

A quadrilateral has vertices $J(0, 0)$ , $K(3, 4)$ , $L(7, 1)$ , $M(4, -3)$ . A student claims the figure is a rhombus.

Compute $JK^2$ . Then compute $KL^2$ . If both squared distances are equal, the student's claim is supported. Enter the value of $JK^2$ .

3.

A circle is centered at $(2, -1)$ and passes through the point $(6, -1)$ .

Determine whether the point $(2, 3)$ lies on the circle. Enter 1 if it does, or 0 if it does not.

E

Find the Mistake

Each problem shows a student's reasoning that contains an error. Identify and explain the mistake.

1.

Alex was given the four points $A(0, 0)$ , $B(4, 0)$ , $C(4, 3)$ , $D(0, 3)$ and asked to prove the quadrilateral is a rectangle.

Alex's work:

Slope of $AB = 0$ , slope of $CD = 0$ → $AB \parallel CD$
Slope of $BC$ is undefined, slope of $DA$ is undefined → $BC \parallel DA$
Conclusion: Since both pairs of opposite sides are parallel, $ABCD$ is a rectangle. ✓

What error did Alex make in this coordinate proof?

A.

Alex computed the slopes of the wrong pairs of sides.

B.

Alex proved the figure is a parallelogram but did not verify perpendicular adjacent sides (right angles), which is required for a rectangle.

C.

Alex should have computed distances instead of slopes to identify a rectangle.

D.

Alex's slope computations are incorrect.

2.

Jordan computed: $AB = \sqrt{(5-1)^2 + (4-1)^2} = \sqrt{25} = 5$ and $CD = \sqrt{(8-4)^2 + (7-4)^2} = \sqrt{25} = 5$ .

Jordan's conclusion: "Since $AB = CD = 5$ , segment $AB$ is parallel to segment $CD$ ."

What error did Jordan make?

A.

Jordan's distance calculations are arithmetic errors.

B.

Jordan confused congruence with parallelism — equal length does not imply parallel direction.

C.

Jordan should have used the midpoint formula, not the distance formula.

D.

Jordan needs to verify perpendicularity, not parallelism.

F

Challenge Problems

These problems require multi-step proofs and careful algebraic reasoning.

1.

Prove or disprove: the quadrilateral with vertices $W(0, 0)$ , $X(5, 0)$ , $Y(6, 4)$ , $Z(1, 4)$ is a parallelogram but not a rectangle. Show all distance and slope computations and state each conclusion with justification.

2.

Using general variable coordinates, place a rhombus with one vertex at the origin and one diagonal along the positive $x$ -axis. Specifically, let the vertices be $A(0, 0)$ , $B(a, b)$ , $C(2a, 0)$ , $D(a, -b)$ where $a > 0$ and $b > 0$ .