G10 Solve problems using slope, distance, and midpoint

Ontario Grade 10 Mathematics

Analytic Geometry

Use coordinates to describe direction, length, and the point halfway between two locations.

A coordinate grid lets us describe where points are and how they are related. You already know how to plot points using an ordered pair: the first number tells the horizontal position, and the second tells the vertical position. In this lesson, you will use those coordinates to answer three different questions. How steep is the segment between two points? How long is it? Where is its halfway point? Each question uses the same two endpoints, but a different calculation.

What you will learn

Find the horizontal and vertical changes between two points on a coordinate grid.
Calculate and interpret the slope between two points.
Find the distance between two points using the distance formula.
Find the midpoint of a line segment.
Choose slope, distance, or midpoint to solve a coordinate problem.

1. From coordinate changes to slope

An ordered pair is written as

(x,y)

. The

x

-coordinate tells how far left or right a point is. The

y

-coordinate tells how far down or up it is. For example, moving from

(1,2)

to

(4,6)

means moving

3

units right and

4

units up.

Slope describes how much a line rises or falls as it moves horizontally. To calculate it, compare the vertical change with the horizontal change. The change in

y

is often called the rise. The change in

x

is often called the run. Always subtract the coordinates in the same order: if you subtract the first point's

x

-coordinate from the second point's, do the same for the

y

-coordinates.

A positive slope means the line rises as you move from left to right. A negative slope means it falls. A horizontal segment has slope

0

, because its vertical change is

0

. A vertical segment has no defined slope: its horizontal change is

0

, so the slope calculation would require division by zero.

Slope can help solve practical coordinate problems. For example, if two locations are on a straight path, slope can describe how much the path rises for each unit of horizontal travel. A steep path has a greater slope magnitude than a shallow path; the magnitude means the distance of the slope value from zero.

m=\frac{y_2-y_1}{x_2-x_1}

Find vertical change and horizontal change using the same endpoint order.
Slope compares vertical change with horizontal change.
A vertical segment has no defined slope because its horizontal change is zero.

2. Distance between two points

Distance answers a different question from slope. It tells the straight-line length of the segment joining two points. If the segment is horizontal or vertical, its length is the change in the coordinate that moves. For a diagonal segment, use the horizontal and vertical changes as the sides of a right triangle.

The Pythagorean relationship says that for a right triangle, the square of the longest side equals the sum of the squares of the other two sides. The diagonal between the points is the longest side of this right triangle. Using the coordinate changes gives the distance formula. The square root makes the answer a length rather than a squared length.

For instance, the points

(1,2)

and

(4,6)

differ by

3

horizontally and

4

vertically. Their connecting segment has length

5

, since

3^2+4^2=25

and the square root of

25

is

5

. Distance is never negative, even when a coordinate change is negative, because the changes are squared.

Use the distance formula when a problem asks how far apart points are, how long a segment is, or whether two segments have equal lengths. Keep the point order consistent in both coordinate differences. Reversing the order changes both signs, but the squared values—and therefore the distance—stay the same.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Distance is the straight-line length between two points.
Square each coordinate change, add the results, and take the square root.
A distance can be left as a square root when it does not simplify to a whole number.

3. Midpoint and choosing a method

The midpoint is the point exactly halfway between two endpoints of a segment. To find it, average the

x

-coordinates to get the midpoint's horizontal position, then average the

y

-coordinates to get its vertical position. Averaging means adding two values and dividing by

2

.

Suppose one endpoint has

x

-coordinate

2

and the other has

x

-coordinate

8

. Halfway between them is

5

. The same idea works for the

y

-coordinates. The result is a point, so write it as an ordered pair.

The three methods answer distinct questions. Use slope for steepness or a rate of vertical change per horizontal change. Use distance for straight-line length. Use midpoint for the halfway location. A problem may ask for more than one of these, so identify what the question is asking before calculating.

A useful way to check a midpoint is to see whether its coordinates lie between the corresponding endpoint coordinates. A useful check for distance is that the answer is non-negative. For slope, check whether the sign matches the line's direction: rising to the right gives a positive slope, and falling to the right gives a negative slope.

M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Average the two $x$ -coordinates and the two $y$ -coordinates separately.
A midpoint is written as an ordered pair.
Choose the calculation that matches the quantity the problem asks for.

4. Guided example and independent practice

Work through the example by first identifying what information is given and what each requested quantity means. Use the same endpoints throughout. The slope describes the segment's direction and steepness, the distance gives its length, and the midpoint gives its halfway location.

After the example, try the questions in the quick check without looking at the answers first. For each one, decide whether it asks for slope, distance, or midpoint. Then select the matching rule and substitute carefully.

Name the quantity before choosing a formula.
Substitute both points consistently and simplify the result.
Use the meaning of the answer to check whether it is reasonable.

What each coordinate method tells you

Question	Use	What the answer represents
How steep is the segment?	Slope	Vertical change for each horizontal change
How long is the segment?	Distance	Straight-line length
Where is the halfway point?	Midpoint	A coordinate pair between the endpoints

Worked example

Three measurements from the same endpoints

Points

A(2,1)

and

B(8,9)

are endpoints of a segment. Find its slope, its length, and its midpoint.

Find the coordinate changes
Use $A$ as the first point and $B$ as the second point. The horizontal change is $8-2=6$ , and the vertical change is $9-1=8$ . Keeping the order consistent helps prevent sign errors.
$\Delta x=6,\quad \Delta y=8$
Calculate the slope
Divide the vertical change by the horizontal change. Both changes are positive, so the segment rises as it moves to the right.
$m=\frac{8}{6}=\frac{4}{3}$
Calculate the length
Square both coordinate changes, add them, and take the square root. The result is the straight-line distance between the endpoints.
$d=\sqrt{6^2+8^2}=\sqrt{100}=10$
Find the midpoint
Average the two $x$ -coordinates and average the two $y$ -coordinates. This gives the point halfway along the segment.
$M=\left(\frac{2+8}{2},\frac{1+9}{2}\right)=(5,5)$

Answer: The slope is

\frac{4}{3}

, the length is

10

units, and the midpoint is

(5,5)

.

Check: The midpoint coordinates are halfway between the corresponding endpoint coordinates. Also, the segment rises to the right, which agrees with its positive slope.

Common mistakes and how to avoid them

Subtracting the

x

-coordinates in one order and the

y

-coordinates in the opposite order when finding slope.

Correction: Choose an endpoint order and use it for both differences. For example, calculate second point minus first point for both coordinates.

Calling the slope of a vertical segment zero.

Correction: A horizontal segment has slope zero. A vertical segment has no defined slope because its horizontal change is zero.

Stopping at the sum of the squared coordinate changes when finding distance.

Correction: Take the square root after adding the squares. The distance is a length, not the squared length.

Adding the endpoint coordinates without dividing by two when finding a midpoint.

Correction: Average each pair of matching coordinates by dividing its sum by two.

Lesson summary

Slope compares vertical change with horizontal change.
Distance is the square root of the sum of the squared coordinate changes.
Midpoint is found by averaging the $x$ -coordinates and the $y$ -coordinates separately.
Read the problem carefully to choose the method that answers its question.

Check your understanding

Question 1

What is the slope between

(1,2)

and

(5,10)

?

$2$
$\frac{1}{2}$
$8$
Undefined

Show answer and explanation

2

The vertical change is

10-2=8

and the horizontal change is

5-1=4

. Their ratio is

8/4=2

.

Question 2

What is the distance between

(0,0)

and

(5,12)

?

$7$ units
$13$ units
$17$ units
$169$ units

Show answer and explanation

13

units

The distance is

\sqrt{5^2+12^2}=\sqrt{169}=13

units.

Question 3

What is the midpoint of

(-2,4)

and

(6,8)

?

$(2,6)$
$(4,12)$
$(2,4)$
$(4,6)$

Show answer and explanation

(2,6)

The averages are

(-2+6)/2=2

and

(4+8)/2=6

, so the midpoint is

(2,6)

.

Key terms

Coordinate: A number that gives a point's position along one direction on a coordinate grid.
Slope: A measure comparing the vertical change of a segment with its horizontal change.
Distance: The straight-line length between two points.
Midpoint: The point exactly halfway between two endpoints.
Endpoint: One of the two points at the ends of a segment.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G10. It is a study resource, not an official curriculum publication.

Official curriculum reference

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G10 · Solve problems using slope, distance, and midpoint

Ontario Grade 10 Mathematics

Use coordinates to describe direction, length, and the point halfway between two locations.

What you will learn

1. From coordinate changes to slope

2. Distance between two points

3. Midpoint and choosing a method

4. Guided example and independent practice

What each coordinate method tells you

Three measurements from the same endpoints

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MPM2D

About this lesson