Back to Exercise: Use coordinates to prove theorems

Exercises: Use Coordinates to Prove Simple Geometric Theorems Algebraically

Work through each section in order. Show all formulas and computations.

Grade 10·21 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-gpe-b-4

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Warm-Up: Review What You Know

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

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

$(a, b)$

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

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

$(2a, 2b)$

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

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

$\dfrac{3}{4}$

$-\dfrac{3}{4}$

$-\dfrac{4}{3}$

$\dfrac{4}{3}$

Fluency Practice

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

$Coordinate grid showing rectangle ABCD with vertices A(0,0), B(4,0), C(4,3), D(0,3) and all four sides labeled$

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 $AB$ .

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 $BC$ .

$Coordinate grid showing quadrilateral PQRS with vertices P(-3,-1), Q(1,-1), R(3,3), S(-1,3)$

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.

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.

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.

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