12.2: Distances in the Coordinate Plane
This activity is intended to supplement Algebra I, Chapter 11, Lesson 5.
ID: 8685
Time required: 40 minutes
Topic: Points, Lines & Planes
- Given the coordinates of the ends of a line segment, calculate its length.
Activity Overview
In this activity, students will explore distances in the coordinate plane. After finding the coordinates of a segment’s endpoints, students will substitute these values into the distance formula and compare the results to the measured length of the segment. Then students will find the distance between the endpoints using the Pythagorean Theorem.
Teacher Preparation
- This activity is designed to be used in a high school or middle school geometry classroom.
- The Distance Formula for the distance between two points \begin{align*}(x_1, \ y_1)\end{align*} and \begin{align*}(x_2, \ y_2)\end{align*} is \begin{align*}\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\end{align*}.
- The Pythagorean Theorem for a right triangle with legs \begin{align*}a\end{align*} and \begin{align*}b\end{align*} and hypotenuse \begin{align*}c\end{align*} is \begin{align*}a^2 + b^2 = c^2\end{align*}. Solving for the hypotenuse, the equation becomes \begin{align*}c=\sqrt{a^2+b^2}\end{align*}.
- Cabri Jr. does not allow the \begin{align*}x-\end{align*}values and \begin{align*}y-\end{align*}values of points to be used separately in calculations. The construction in Problem 1 uses a perpendicular line to project the coordinate on the appropriate axis, and then the distance from \begin{align*}(0, \ 0)\end{align*} is used to find the \begin{align*}x-\end{align*}values and \begin{align*}y-\end{align*}values.
- The screenshots on pages 1–6 demonstrate expected student results.
- To download Cabri Jr. and the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=8685 and select Cabri Jr., DISTNC1.8xv, and DSTNC2.8xv.
Associated Materials
- Student Worksheet: Distances in the Coordinate Plane, http://www.ck12.org/flexr/chapter/9621, scroll down to the second activity.
- Cabri Jr.
- DISTNC1, DISTNC2
Classroom Management
- This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.
- The student worksheet GeoWeek04_Distance_Worksheet_TI-84 helps guide students through the activity and provides a place for students to record their answers and observations.
- Depending on student skill level, you may wish to use points with integer coordinates, or only positive values.
- Note: The coordinates can display , \begin{align*}1\end{align*}, or \begin{align*}2\end{align*} decimal digits. If digits are displayed, the value shown will round from the actual value. To ensure that a point is actually at an integer value rather than a rounded decimal value, do the following:
- Move the cursor over the coordinate value so it is highlighted.
- Press + to display additional decimal digits or - to hide digits.
Problem 1 – The Distance Formula
Note: If the file Distncl is distributed to student calculators, skip Steps 1 and 2. Proceed with Step 3.
Step 1: Students should open a new Cabri Jr. file. If the axes are not currently showing, they should select Hide/Show > Axes.
Students are to construct a segment in the first quadrant using the Segment tool.
Step 2: In order to use the \begin{align*}x-\end{align*}values and \begin{align*}y-\end{align*}values of the endpoints separately in calculations, students will use a perpendicular line to project the coordinates on the appropriate axis.
Direct students to select the Perp. tool to construct a line through one endpoint of the segment perpendicular to each axis.
Find the intersection of the perpendicular line with the axis using the Point > Intersection tool. Then, hide the perpendicular line with the Hide/Show > Object tool.
Repeat this process for the second endpoint.
Step 3: Students will select Coord. & Eq. and show the coordinates for the endpoints of the segment.
If the coordinates of the endpoints are not integers, they need to use the Hand tool to drag the endpoints until the coordinates are integers.
Step 4: Students should measure the distance from each axis point to the origin using the Measure > D. & Length tool. These distances should match the \begin{align*}x-\end{align*}values and \begin{align*}y-\end{align*}values of the coordinates.
Students can now drag the endpoints and observe that the axis point distances change as the segment endpoints move, but still match the coordinates.
Step 5: Have students measure the length of the segment using the Measure > D. & Length tool.
Step 6: The Calculate tool can perform calculations on pairs of numbers. The Distance Formula calculations will be broken down into individual steps. Be sure to select coordinates in the proper order!
Students should complete the following steps in order:
a. Subtract the two \begin{align*}x-\end{align*}values.
b. Multiply this result by itself.
c. Subtract the two \begin{align*}y-\end{align*}values.
d. Multiply this result by itself.
e. Add the two products.
f. Square root this sum.
Note: If desired, have students do the calculations directly on the worksheet rather than within the Cabri Jr. file.
Now students can drag the segment endpoints and observe the calculation results as they update.
Discuss how these calculation results relate to the measured length of the segment.
Problem 2 – Apply the Math
Note: If the file Distnc2 is distributed to student calculators, skip Steps 1, 2 and 3. Proceed with Step 4.
Step 1: Students should open a new Cabri Jr. file. If the axes are not currently showing, they should select Hide/Show > Axes.
Students are to construct a segment in the first quadrant using the Segment tool.
Step 2: Students will be constructing a small right triangle for the segment such that the segment is the hypotenuse of the right triangle.
Select the Perp. tool and construct a line through the upper segment endpoint perpendicular to the \begin{align*}x-\end{align*}axis. Construct a second line through the lower segment endpoint perpendicular to the \begin{align*}y-\end{align*}axis.
Have students find the intersection of the perpendicular lines by selecting Point > Intersection.
Step 3: Direct students to hide the perpendicular lines with the Hide/Show > Object tool. Do not hide the intersection point.
They should then construct segments for the legs of the right triangle with the Segment tool.
Step 4: Tell students to find the length of each side of the triangle.
Discuss the lengths of the three sides of the triangle and decide which is longest. Identify the legs and the hypotenuse.
Step 5: Use the Calculate tool to perform the following calculations:
Students should complete the following steps in order:
a. Multiply the length of one leg by itself.
b. Multiply the length of the other leg by itself.
c. Add the two values.
d. Square root this sum.
Note: If desired, have students do the calculations directly on the worksheet rather than within the Cabri Jr. file.
Step 6: Students should show the coordinates of the segment endpoints and calculate the length using the Distance Formula on their worksheet.
Students should drag the endpoints of the segment and observe the relationship between the \begin{align*}3\end{align*} values:
- Measured length
- Calculated length (Distance Formula) on worksheet
- Calculated Pythagorean distance
Discuss the connections between the Distance Formula and the Pythagorean Theorem. Challenge students to explain how \begin{align*}\sqrt{a^2+b^2}\end{align*} is derived from the Pythagorean Theorem.
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