Use Coordinates to Prove Geometric Theorems

The Lesson Activities will help you meet these educational goals:

  • Content Knowledge—You will use coordinates to prove simple geometric propertiesalgebraically, including proofs involving circles.
  • Inquiry—You will perform an investigationin which you will make observations.

Directions

You will evaluatesome of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

  1. Using Coordinates with Triangles

You will use GeoGebra to classifytriangles by measuring their sides and angles. To get started,open GeoGebra and complete each step below. If you need help, follow these instructions for using GeoGebra.

  1. Mark the points (10, 14), (15, 11.11), and (10, 8.22). Enter the coordinates in the input window, if you wish. Then connect the points to form a triangle.
  1. Measure the lengths of the sides of the triangle and record them in the table.

Type your response here:

Side of Triangle / Length
  1. Take a screenshot of the triangle with the lengths labeled, and paste it below. Based on the side lengths, which type of triangle is it?Explain your answer.

Type your response here:

  1. Now open GeoGebraagain.Mark the points with the coordinates (4, 14), (22, 6), and (16, 18). Connect the points to form a triangle.
  1. Measure the lengths of the sides and angles of the triangle, and record them in the table.

Type your response here:

Side of Triangle / Length / Angle of Triangle / Measure
  1. Take a screenshot of the triangle with its side lengths and angle measures labeled,and paste it below. Based on the measurements, which type of triangle is it? Explain your answer.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Using Coordinates with Quadrilaterals

You will now use GeoGebrato classifya quadrilateral by measuring the slopes of the sides of the quadrilateral.

  1. Mark the points A(10, 10), B(22, 2), C(30, 8),and D(12, 20), and connect them to form a quadrilateral. Measure the slopes of the sides of the quadrilateral, and note them in the table.

Type your response here:

Side of Quadrilateral / Slope
  1. Take a screenshot of the quadrilateral with its slopeslabeled,and paste it below. Based on the slopes, which type of quadrilateral is it? Explain your answer.

Type your response here:

  1. Revisit your work in part b. Is the trapezoid that you created an isosceles trapezoid? How can you tell? If you didn’t have geometry software, how would you prove or disprove that the trapezoid is isosceles? Describe your procedure.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Using Coordinates with Circles

You will use GeoGebra to determine whether certain points lie on a given circle.

  1. Mark the points A(5, 5) and B(7, 2). Draw a circle that has pointA as the centerand passesthrough point B.Measure and label the length of a radius of the circle.What is the radius of circle A?

Type your response here:

  1. Now graphthe pointsC(2, 8) and D(8, 7). (Try entering the coordinates through the input window.)Measure the lengthsofand . Do the points C and D lie on the circle?How do you know? Take a screenshot showing the points and their distances from the center, and paste it below.

Type your response here:

How did you do? Check a box below.

Nailed It!—Iincludedall of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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