Lehman PTS-3 Summer 2020

1. 𝐏𝐓𝐒 𝟑
Bridge to Calculus Workshop
Summer 2020
Lesson 13
Midpoint and Distance
Formulas
“Instead of concentrating just on
finding good answers to questions,
it's more important to learn how to
find good questions!”
- Donald E. Knuth-

2. Lehman College, Department of Mathematics
Midpoint Formula (1 of 4)
The midpoint 𝑥 of the line segment joining two points 𝑥1
and 𝑥2 on a number line is given by:
Example 1. Find the midpoint 𝑥 of the line segment
joining the points 𝑥 = 3 and 𝑥 = −1 on the number line.
Solution. Using the midpoint formula:
𝑥 =
𝑥1 + 𝑥2
2
𝑥 =
𝑥1 + 𝑥2
2
=
3 + (−1)
2
=
2
2
= 1

3. Lehman College, Department of Mathematics
Midpoint Formula (2 of 4)
Let 𝑥1 and 𝑥2 be two points on a number line. Without
loss of generality, let 𝑥1 < 𝑥2.
Now, Let 𝑥 be the midpoint of the line segment [𝑥1, 𝑥2].
It follows that the distance from 𝑥1 to 𝑥 must equal the
distance from 𝑥 to 𝑥2. That is:
𝑥 − 𝑥1 = 𝑥2 − 𝑥
𝑥 − 𝑥1 + 𝑥 = 𝑥2 − 𝑥 + 𝑥
2𝑥 − 𝑥1 = 𝑥2
2𝑥 − 𝑥1 + 𝑥1 = 𝑥2 + 𝑥1

4. Lehman College, Department of Mathematics
Midpoint Formula (3 of 4)
From the previous slide:
In the coordinate plane, the midpoint (𝑥, 𝑦) of the line
segment joining the points (𝑥1, 𝑦1) and (𝑥2, 𝑦2) is given
by the formula:
Example 2. Find the midpoint 𝑥, 𝑦 of the line segment
joining the points (−2, 3) and (6, −5) in the plane.
2𝑥 − 𝑥1 + 𝑥1 = 𝑥2 + 𝑥1
2𝑥 = 𝑥1 + 𝑥2
𝑥 =
𝑥1 + 𝑥2
2
𝑥, 𝑦 =
𝑥1 + 𝑥2
2
,
𝑦1 + 𝑦2
2

5. Lehman College, Department of Mathematics
Midpoint Formula (4 of 4)
Example 2. Find the midpoint 𝑥, 𝑦 of the line segment
joining the points (−2, 3) and (6, −5) in the plane.
Solution. Using the midpoint formula:
𝑥, 𝑦 =
𝑥1 + 𝑥2
2
,
𝑦1 + 𝑦2
2
=
−2 + 6
2
,
3 + (−5)
2
=
4
2
,
−2
2
= 2, −1

6. Lehman College, Department of Mathematics
Distance Formula (1 of 4)
The distance 𝑑 between two points 𝑥1 and 𝑥2 on a
number line is given by:
Example 3. Determine the distance 𝑑 between the
points 𝑥 = −3 and 𝑥 = −7 on a number line.
Solution. Use the distance formula:
𝑑 = | 𝑥2 − 𝑥1|
𝑑 = | 𝑥2 − 𝑥1|
= −3 − (−7) = −3 + 7 = 4 = 4

7. Lehman College, Department of Mathematics
Distance Formula (2 of 4)
Let 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2) be two points in the plane.
Denote the distance between 𝐴 and 𝐵 by 𝑑 = 𝑑 𝐴, 𝐵 .
Construct horizontal and vertical lines to meet at 𝐶.
Determine the lengths of the legs of right triangle 𝐴𝐵𝐶.

8. Lehman College, Department of Mathematics
Distance Formula (3 of 4)
How do we determine the distance 𝑑(𝐴, 𝐵)?
Use the Pythagorean Theorem to determine 𝑑(𝐴, 𝐵):
𝑑2
= 𝑥2 − 𝑥1
2
+ 𝑦2 − 𝑦1
2
𝑑(𝐴, 𝐵) = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2

9. Lehman College, Department of Mathematics
Distance Formula (4 of 4)
Example 4. Find the distance between the points
𝐴(2, 5) and 𝐵 4, −1 in the coordinate plane.
Solution. Using the distance formula, we have:
𝑑(𝐴, 𝐵) = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
= 4 − 2 2 + −1 − 5 2
= 2 2 + −6 2
= 4 + 36 = 40
= 4 ⋅ 10 = 4 ⋅ 10
= 2 10

10. Lehman College, Department of Mathematics
Distance Formula (4 of 4)
Example 5. Find the distance between the points
𝐴(−5, 2) and 𝐵 −1, −2 in the coordinate plane.
Solution. Using the distance formula, we have:
𝑑(𝐴, 𝐵) = 𝑥2 − 𝑥1
2 + 𝑦2 − 𝑦1
2
= −5 − (−1) 2 + 2 − (−2) 2
= −4 2 + 4 2
= 16 + 16 = 32
= 16 ⋅ 2 = 16 ⋅ 2
= 4 2