Topics
7. Angles & Triangles
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United States of America
Grade 8

7.05 Distances in the plane

Lesson
Practice

Distances in the plane

We already learned how to use  Pythagorean theorem  to calculate the side lengths in a right triangle. Pythagorean' theorem states:a^2+b^2=c^2

Did you know we can also use the Pythagorean theorem to find the distance between two points on a coordinate plane? Let's look at an example.

Examples

Example 1

How far is the point P(-15,8) from the origin?

Worked Solution
Create a strategy

Draw a right triangle with vertices at P and the origin and use Pythagorean theorem to calculate the distance.

Apply the idea

In the right triangle below, the shorter side lengths are a=8 units and b=15 units.

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\displaystyle c^2\displaystyle =\displaystyle a^2 + b^2Use the Pythagorean theorem
\displaystyle =\displaystyle 8^2 + 15^2Substitute the values
\displaystyle =\displaystyle 64 + 225Evaluate the squares
\displaystyle =\displaystyle 289 Evaluate
\displaystyle \sqrt{c^2}\displaystyle =\displaystyle \sqrt{289}Square root both sides
\displaystyle c\displaystyle =\displaystyle 17 Evaluate the square root

So, the distance between P and the origin is 17\,units.

Example 2

The points A(-3,-2), \, B(-3,-4)and C(1,-4) are the vertices of a right triangle, as shown on the number plane.

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a

Find the length of interval \text{AB}.

Worked Solution
Create a strategy

The length of the interval is the difference between the y-coordinates of the two points.

Apply the idea
\displaystyle AB\displaystyle =\displaystyle -2 - (- 4)Subtract the y-coordinates
\displaystyle =\displaystyle 2 \text{ units}Evaluate
b

Find the length of interval \text{BC}.

Worked Solution
Create a strategy

The length of the interval is the difference between the x-coordinates of the two points.

Apply the idea
\displaystyle BC\displaystyle =\displaystyle 1 - (- 3)Subtract the x-coordinates
\displaystyle =\displaystyle 4 \text{ units}Evaluate
c

If the length of \text{AC} is denoted by c, use Pythagoras’ theorem to find the value of c to three decimal places.

Worked Solution
Create a strategy

Use the Pythagorean theorem: a^2+b^2=c^2.

Apply the idea

We have found that\,a = 2 and b = 4.

\displaystyle c^2\displaystyle =\displaystyle a^2 + b^2Use the Pythagorean theorem
\displaystyle =\displaystyle 2^2 + 4^2Substitute the values
\displaystyle =\displaystyle 4 + 16Evaluate the squares
\displaystyle =\displaystyle 20 Evaluate
\displaystyle \sqrt{c^2}\displaystyle =\displaystyle \sqrt{20}Square root both sides
\displaystyle c\displaystyle =\displaystyle 4.472Evaluate to three decimal places

So, the length of AC is 4.472\,units.

Idea summary
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We can use the Pythagorean theorem to find the distance between two points on a coordinate plane by drawing a right triangle with vertices at the two points.

By letting c be the distance between the two points, we can find this length using the formula:c^2=a^2+b^2

Outcomes

8.G.B.8

Apply the Pythagorean theorem to find the distance between two points in a coordinate system.

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