G.7(C)·Geometry·Coordinate GeometryReadiness

Distance, Midpoint, and Slope

Use distance, midpoint, and slope formulas to find lengths and points on lines.

Students use midpoint, distance, and slope on the coordinate plane, including finding missing endpoints, locating points on a line with a given slope, and writing distance expressions with variable coordinates.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Geometry released practice55 minutes

Learn the concept

Midpoints average coordinates

The midpoint of the segment from $(x_1,y_1)$ to $(x_2,y_2)$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ . It is the point halfway along the segment in both directions.

Finding a missing endpoint

From the known endpoint to the midpoint, note the horizontal and vertical change. Apply the same change again from the midpoint to reach the other endpoint.

Distance from a right triangle

The distance between two points is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ . The horizontal and vertical changes are the legs of a right triangle, and the segment is its hypotenuse.

Slope and another point on a line

Slope measures vertical change divided by horizontal change: $m=\frac{y_2-y_1}{x_2-x_1}$ . Subtract coordinates in the same order. Through $(2,7)$ with slope $-\frac{3}{2}$ , move right 2 and down 3 to reach $(4,4)$ . More generally, $y-y_1=m(x-x_1)$ . A vertical line has undefined slope because its horizontal change is zero.

Distance with variable coordinates

Variables work just like numbers in the distance formula. From $R(u,-3)$ to $S(2,v)$ , the horizontal change is $2-u$ and the vertical change is $v-(-3)=v+3$ . Thus $RS=\sqrt{(2-u)^2+(v+3)^2}$ . Keep each entire difference inside parentheses before squaring.

Worked example

Segment AB has endpoint A(−3, 4) and midpoint M(1, 1). Find B and the length of AB.

From A to M, x changes by +4 and y changes by −3.
Apply the same change from M: B = (1 + 4, 1 − 3) = (5, −2).
$AM=\sqrt{4^2+(-3)^2}=5$ .
AB is twice AM.

B(5, −2); AB = 10 units

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Geometry - Using the Midpoint Formula

Five Minute Math · Aaron Daffern, Dallas ISD

Find midpoints and missing endpoints with the midpoint formula.

Watch on YouTube

Texas public school video

Geometry - Deriving the Distance Formula

Five Minute Math · Aaron Daffern, Dallas ISD

Derive the distance formula from the Pythagorean theorem on the coordinate plane.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Find the midpoint of the segment from (−2, 7) to (6, −1).
Show answer and reasoning
(2, 3)
Average the coordinates: $\frac{-2+6}{2}=2$ and $\frac{7+(-1)}{2}=3$ .
2. Find the distance between (1, 2) and (7, 10).
Show answer and reasoning
10 units
The changes are 6 and 8, so $d=\sqrt{36+64}=\sqrt{100}=10$ .
3. Find the slope through (2, 7) and (6, 1). Then give another point on that line.
Show answer and reasoning
Slope $-\frac{3}{2}$ ; one possible point is $(4,4)$ .
$\frac{1-7}{6-2}=-\frac{6}{4}=-\frac{3}{2}$ . From $(2,7)$ , move right 2 and down 3.
4. Write the distance from $A(t,-5)$ to $B(4,w)$ without assigning values to t or w.
Show answer and reasoning
$\sqrt{(4-t)^2+(w+5)^2}$
The coordinate changes are $4-t$ and $w-(-5)=w+5$ . Square both changes, add, and take the square root.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 1 · 1 point

$\overline{CD}$ has an endpoint at $(2, -1)$ and a midpoint at $(8, 3)$ . Which measure is closest to the length of $\overline{CD}$ ?

20.4 units
8.9 units
14.4 units
11.7 units

Show STAAR answer and explanation

14.4 units

The other endpoint is $(2(8)-2,2(3)-(-1))=(14,7)$ . Thus $CD=\sqrt{(14-2)^2+(7-(-1))^2}=\sqrt{208}\approx14.4$ units.

20.4 units: This overestimates the distance. The horizontal and vertical changes between the endpoints are 12 and 8, giving $\sqrt{208}$ .

8.9 units: This is too short for a segment whose horizontal change alone is 12 units.

11.7 units: This does not equal $\sqrt{12^2+8^2}$ ; the full length is approximately 14.4 units.

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Archived released STAAR Geometry EOC · 2013 · Item 29 · 1 point

The slope of a line passing through $H(-2, 5)$ is $-\frac{3}{4}$ . Which ordered pair represents a point on this line?

(6, −1)
(2, 8)
(−5, 1)
(1, 1)

Show STAAR answer and explanation

(6, −1)

From (−2, 5) to (6, −1), the changes are 8 horizontally and −6 vertically. The slope is $-6/8=-3/4$ .

(2, 8): The slope to (2, 8) is $3/4$ , with the wrong sign.

(−5, 1): The slope to (−5, 1) is $(-4)/(-3)=4/3$ , not −3/4.

(1, 1): The slope to (1, 1) is −4/3, the reciprocal of the required slope.

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Archived released STAAR Geometry EOC · 2013 · Item 44 · 1 point

A civil engineer is drawing a plan for the location and length of a new underground sewer pipe on a coordinate grid. The pipe on the plan will run from point $N(a, -2)$ to point $P(1, b)$ on the coordinate grid. Which expression represents the shortest distance between $N$ and $P$ in units?

$(a + 2)^2 + (1 - b)^2$
$(1 - a)^2 + (b + 2)^2$
$\sqrt{(a + 2)^2 + (1 - b)^2}$
$\sqrt{(1 - a)^2 + (b + 2)^2}$

Show STAAR answer and explanation

$\sqrt{(1 - a)^2 + (b + 2)^2}$

The horizontal difference is $1-a$ and the vertical difference is $b-(-2)=b+2$ . By the distance formula the length is $\sqrt{(1-a)^2+(b+2)^2}$ .

$(a + 2)^2 + (1 - b)^2$ : This mixes x- and y-coordinates and omits the square root.

$(1 - a)^2 + (b + 2)^2$ : This gives squared distance, not distance; take the square root.

$\sqrt{(a + 2)^2 + (1 - b)^2}$ : This mixes coordinates from different axes instead of subtracting corresponding coordinates.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Find a midpoint by averaging the endpoint coordinates.
Use a midpoint and one endpoint to locate the other endpoint.
Use the distance formula with numerical or variable coordinates to find segment lengths.
Calculate slope from two points and use a given slope to locate another point on a line.

Instructional sequence

1
Engage
5 min
Plot a segment and its midpoint. Ask how to find the other end if only one endpoint and the midpoint are known.
2
Direct instruction
10 min
Derive the midpoint formula as an average and the distance formula from a right triangle drawn on the grid.
3
Guided practice
12 min
Find midpoints, then use “same change again” to locate missing endpoints.
4
Length problems
12 min
Find full segment lengths by computing the endpoint-to-endpoint distance or doubling the half-length.
5
Slope and variable coordinates
10 min
Compute slopes, use rise and run to locate another point, and write distance expressions without assigning values to variables.
6
Closure
6 min
Students choose the most efficient path in the exit ticket and explain it.

Misconceptions to address

““The midpoint formula subtracts the coordinates.””

Teacher move: A midpoint is the average position, so add the coordinates and divide by 2. Subtraction belongs in the distance formula.

““The distance to the midpoint is the segment's length.””

Teacher move: That distance is only half of the segment. Double it or find the other endpoint first.

““A negative coordinate stays negative when I subtract it.””

Teacher move: Subtracting a negative adds: $v-(-3)=v+3$ . Write the subtraction with parentheses before simplifying.

Materials

•Coordinate grids
•Segment cards with labeled endpoints
•Calculator

Teaching tips

Draw the right triangle on the grid before using the distance formula.
Use arrows labeled with the horizontal and vertical change to find a missing endpoint.
Ask whether the answer is a whole segment or half of one before rounding.

Exit ticket · 3–5 min

A segment has an endpoint at (0, 0) and a midpoint at (3, 4). What is the segment's length?

Show expected answer

The endpoint-to-midpoint distance is $\sqrt{3^2+4^2}=5$ , so the full segment is 10 units.

Sources & alignment

TEA 2013 Geometry released test TEA 2013 Geometry answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Midpoints average coordinates

Finding a missing endpoint

Distance from a right triangle

Slope and another point on a line

Distance with variable coordinates

Worked example

Watch the lesson

Geometry - Using the Midpoint Formula

Geometry - Deriving the Distance Formula

Try it yourself

1. Find the midpoint of the segment from (−2, 7) to (6, −1).

2. Find the distance between (1, 2) and (7, 10).

3. Find the slope through (2, 7) and (6, 1). Then give another point on that line.

4. Write the distance from A(t,−5)A(t,-5)A(t,−5) to B(4,w)B(4,w)B(4,w) without assigning values to t or w.

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 1 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 29 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 44 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Length problems

Slope and variable coordinates

Closure

Misconceptions to address

4. Write the distance from $A(t,-5)$ to $B(4,w)$ without assigning values to t or w.