Objectives
- Given two points, determine the distance between them and the midpoint of the line segment connecting them.
Section 1.3 Distance and Midpoint (EQ3)
Subsection 1.3.1 Activities
Activity 1.3.1.
The points \(A \) and \(B \) are shown in the graph below. Use the graph to answer the following questions:
(a)
Draw a right triangle so that the hypotenuse is the line segment between points \(A\) and \(B \text{.}\) Label the third point of the triangle \(C\text{.}\)
Answer.
Point \(C\) should be at the point \((2,2)\text{.}\)
(b)
Find the lengths of line segments \(AC \) and \(BC \text{.}\)
Answer.
\(AC\) is \(4\) units long. \(BC\) is \(2\) units long. Make sure students pay attention to the scale.
(c)
Now that you know the lengths of \(AC \) and \(BC \text{,}\) how can you find the length of \(AB \text{?}\) Find the length of \(AB\text{.}\)
Answer.
Students should see that they can use the Pythagorean Theorem to find the length of side \(AB\) (which is \(\sqrt{20}\) or approximately \(4.5\)).
Remark 1.3.3.
Using the Pythagorean Theorem \((a^2+b^2=c^2)\) can be helpful in finding the distance of a line segment (as long as you create a right triangle!).
Activity 1.3.4.
Suppose you are given two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\text{.}\) Let’s investigate how to find the length of the line segment that connects these two points!
(a)
Draw a sketch of a right triangle so that the hypotenuse is the line segment between the two points.
Answer.
Students may need help in their drawing. Make sure the hypotenuse is the line segment that connects the two points.
(b)
Find the lengths of the legs of the right triangle.
Answer.
The lengths of the legs of the triangle should be \(y_{2}-y_{1}\) and \(x_{2}-x_{1}\) (or \(y_{1}-y_{2}\) and \(x_{1}-x_{2}\) depending on how students created their drawing).
(c)
Find the length of the line segment that connects the two original points.
Answer.
Students should see the connection to the previous activity and apply the Pythagorean Theorem. They should get either \(\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}\) or \(\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2}\) to represent the length of the side that connects the two points.
Definition 1.3.5.
The distance, \(d\text{,}\) between two points, \((x_{1},y_{1})\) and \((x_{2}, y_{2})\text{,}\) can be found by using the distance formula:
\begin{equation*}
d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}
\end{equation*}
Notice that the distance formula is an application of the Pythagorean Theorem!
Activity 1.3.6.
Apply Definition 1.3.5 to calculate the distance between the given points.
(a)
What is the distance between \((4,6)\) and \((9,15)\text{?}\)
- \(\displaystyle 10.2\)
- \(\displaystyle 10.3\)
- \(\displaystyle \sqrt{106}\)
- \(\displaystyle \sqrt{56}\)
Answer.
B and C
(b)
What is the distance between \((-2,5)\) and \((-7,-1)\text{?}\)
- \(\displaystyle \sqrt{11}\)
- \(\displaystyle 7.8\)
- \(\displaystyle 3.3\)
- \(\displaystyle \sqrt{61}\)
Answer.
B and D
(c)
Suppose the line segment \(AB\) has one endpoint, \(A\text{,}\) at the origin. For which coordinate of \(B\) would make the line segment \(AB\) the longest?
- \(\displaystyle (3,7)\)
- \(\displaystyle (2,-8)\)
- \(\displaystyle (-6,4)\)
- \(\displaystyle (-5,-5)\)
Answer.
B
Remark 1.3.7.
Notice in Activity 1.3.6, you can give a distance in either exact form (leaving it with a square root) or as an approximation (as a decimal). Make sure you can give either form as sometimes one form is more useful than another!
Remark 1.3.8.
A midpoint refers to the point that is located in the middle of a line segment. In other words, the midpoint is the point that is halfway between the two endpoints of a given line segment.
Activity 1.3.9.
Two line segments are shown in the graph below. Use the graph to answer the following questions:
(a)
What is the midpoint of the line segment \(AB\text{?}\)
- \(\displaystyle (16,4)\)
- \(\displaystyle (8,4)\)
- \(\displaystyle (8,8)\)
- \(\displaystyle (10,2)\)
Answer.
B
(b)
What is the midpoint of the line segment \(AC\text{?}\)
- \(\displaystyle (6,0)\)
- \(\displaystyle (4,4)\)
- \(\displaystyle (6,4)\)
- \(\displaystyle (5,2)\)
Answer.
D
(c)
Suppose we connect the two endpoints of the two line segments together, to create the new line segment, \(BC\text{.}\) Can you make an educated guess to where the midpoint of \(BC\) is?
- \(\displaystyle (10,8)\)
- \(\displaystyle (6,4)\)
- \(\displaystyle (5,4)\)
- \(\displaystyle (5,2)\)
Answer.
C
(d)
How can you test your conjecture? Is there a mathematical way to find the midpoint of any line segment?
Definition 1.3.11.
The midpoint of a line segment with endpoints \((x_{1},y_{1})\) and \((x_{2}, y_{2})\text{,}\) can be found by taking the average of the \(x\) and \(y\) values. Mathematically, the midpoint formula states that the midpoint of a line segment can be found by:
\begin{equation*}
\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)
\end{equation*}
Activity 1.3.12.
Apply Definition 1.3.11 to calculate the midpoint of the following line segments.
(a)
What is the midpoint of the line segment with endpoints \((-4,5)\) and \((-2,-3)\text{?}\)
- \(\displaystyle (3,1)\)
- \(\displaystyle (-3,1)\)
- \(\displaystyle (1,1)\)
- \(\displaystyle (1,4)\)
Answer.
B
(b)
What is the midpoint of the line segment with endpoints \((2,6)\) and \((-6,-8)\text{?}\)
- \(\displaystyle (-3,-1)\)
- \(\displaystyle (-2,0)\)
- \(\displaystyle (-2,-1)\)
- \(\displaystyle (4,7)\)
Answer.
C
(c)
Suppose \(C\) is the midpoint of \(AB\) and is located at \((9,8)\text{.}\) The coordinates of \(A\) are \((10,10)\text{.}\) What are the coordinates of \(B\text{?}\)
- \(\displaystyle (9.5,9)\)
- \(\displaystyle (11,12)\)
- \(\displaystyle (18,16)\)
- \(\displaystyle (8,6)\)
Answer.
D
Activity 1.3.13.
On a map, your friend Sarah’s house is located at \((-2, 5)\) and your other friend Austin’s house is at \((6,-2)\text{.}\)
(a)
How long is the direct path from Sarah’s house to Austin’s house?
Answer.
\(\sqrt{113}\) or \(10.6\)
(b)
Suppose your other friend, Micah, lives in the middle between Sarah and Austin. What is the location of Micah’s house on the map?
Answer.
\((2,1.5)\)
Subsection 1.3.2 Videos
It would be great to include videos down here, like in the Calculus book!
