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9.1: The Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 8, Lesson 1.

ID: 9532

Time required: 60 minutes

Activity Overview

In this activity, students will use the Cabri Jr. application to construct figures that prove the Pythagorean Theorem in two different ways.

Topic: Right Triangles & Trigonometric Ratios

• Construct and measure the side lengths of several right triangles and conjecture a relationship between the areas of the squares drawn on each side.
• Prove and apply the Pythagorean Theorem.

Teacher Preparation

• This activity is designed to be used in a high school or middle school geometry classroom.
• 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. Feel free to print out the following pages for your students.
• The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. This can be expressed as \begin{align*}c^2 = a^2 + b^2\end{align*} where \begin{align*}c\end{align*} is the length of the hypotenuse.
• Depending on student skill level, you may wish to download the constructed figures to student calculators. If the files are downloaded, skip the construction steps for each problem and begin each at Step 10.
• Note: Measurements can display 0, 1, or 2 decimal digits. If 0 digits are displayed, the value shown will round from the actual value. To change the number of digits displayed:
1. Move the cursor over the value so it is highlighted.
2. Press + to display additional decimal digits or - to hide digits.

Associated Materials

Problem 1 – Squares on Sides Proof

The Pythagorean Theorem states that, the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. In this activity, you will construct a right triangle and verify the Pythagorean Theorem by constructing squares on each side and comparing the sum of the area of the two smaller squares to the area of the square of the third side.

Step 1: Open a new Cabri Jr. file.

Construct a segment using the Segment tool.

Select the Alph-Num tool to label the endpoints \begin{align*}B\end{align*} and \begin{align*}C\end{align*} as shown.

Step 2: Construct a line through \begin{align*}C\end{align*} that is perpendicular to \begin{align*}\overline{BC}\end{align*} using the Perp. tool.

Step 3: Construct a point on the perpendicular line and label it \begin{align*}A\end{align*}.

Hide the perpendicular line with the Hide/Show tool and construct line segments \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{AB}\end{align*}.

For the time being, keep the sides of the triangle fairly small so that squares can be constructed on the sides.

Step 4: In the lower left corner, use the Alph-Num tool to place the number 90 on the screen. This will be the angle of rotation.

Note: Press ALPHA to access numerical characters. A small “1” will appear in the tool icon in the upper left corner of the screen.

Step 5: Use the Rotation tool to rotate point \begin{align*}C\end{align*} about point \begin{align*}B\end{align*} through an angle of \begin{align*}90^\circ\end{align*}.

• Press ENTER on point \begin{align*}B\end{align*} as the center of rotation.
• Press ENTER on the angle of rotation (the number 90).
• Press ENTER on point \begin{align*}C\end{align*}, the object to be rotated.

Notice that the number now has a degree symbol associated with it and that the point has been rotated in the counter-clockwise direction.

Step 6: What we want to do next is to rotate point \begin{align*}B\end{align*} about point \begin{align*}C\end{align*} through an angle of \begin{align*}90^\circ\end{align*} in the clockwise direction. To do this we will need an angle of -90. Place this number on the screen.

Using the value of -90, rotate point \begin{align*}B\end{align*} about point \begin{align*}C\end{align*} through an angle of \begin{align*}-90^\circ\end{align*}.

Step 7: You should now have two points below the line segment \begin{align*}\overline{BC}\end{align*}. Use the Quad. tool to construct a quadrilateral using points \begin{align*}B\end{align*}, \begin{align*}C\end{align*} and the two points constructed in Steps 5 and 6.

Answer Question 1 on the worksheet.

Step 8: In a similar fashion, rotate point \begin{align*}C\end{align*} about point \begin{align*}A\end{align*} through an angle of \begin{align*}-90^\circ\end{align*} and rotate point \begin{align*}A\end{align*} about point \begin{align*}C\end{align*} through an angle of \begin{align*}90^\circ\end{align*}. This will allow us to construct a second square.

Use the Quad. tool again to construct the square on side \begin{align*}\overline{AC}\end{align*}.

Step 9: Finally, rotate point \begin{align*}B\end{align*} about point \begin{align*}A\end{align*} through an angle of \begin{align*}90^\circ\end{align*} and rotate point \begin{align*}A\end{align*} about point \begin{align*}B\end{align*} through an angle of \begin{align*}-90^\circ\end{align*}.

Then construct a third square on hypotenuse \begin{align*}\overline{AB}\end{align*}.

Step 10: Start with this step if you are using the pre-constructed file “PYTHAG1”.

Select the Measure > Area tool and measure the area of the three squares.

Step 11: Using the Calculate tool, press ENTER on the measurements of the two smaller squares and then press the + key. Place the sum off to the side of the screen.

How does this sum compare to the square of the hypotenuse? Record your observations in the table for Question 2 on the worksheet.

Step 12: To test your construction, drag points \begin{align*}A\end{align*}, \begin{align*}B\end{align*} and/or \begin{align*}C\end{align*} to a new location on the screen.

Answer Question 3 on the worksheet.

Problem 2 – Inside a Square Proof

In this problem, we are going to look at a proof of the Pythagorean Theorem. We hope to prove the statement that, if \begin{align*}z\end{align*} is the length of the hypotenuse of a right triangle and \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are the lengths of the legs of the right triangle, then \begin{align*}z^2 = x^2 + y^2\end{align*}.

Step 1: Construct a line segment \begin{align*}\overline{AB}\end{align*}.

Use the Alph-Num tool to place the value \begin{align*}90\end{align*} on your screen.

Step 2: Access the Rotation tool and press ENTER on point \begin{align*}A\end{align*} as the center of the rotation, then on 90 as the angle of rotation and finally on line segment \begin{align*}\overline{AB}\end{align*} as the object to be rotated.

Label the new point \begin{align*}D\end{align*}.

Step 3: Continue by rotating line segment \begin{align*}\overline{AD}\end{align*} about point \begin{align*}D\end{align*} through an angle of \begin{align*}90^\circ\end{align*}.

Label the new point \begin{align*}C\end{align*}.

Step 4: Complete the square by constructing line segment \begin{align*}\overline{BC}\end{align*}.

Step 5: Using the Point on tool, add point \begin{align*}E\end{align*} on \begin{align*}\overline{AB}\end{align*} as shown and overlay a line segment \begin{align*}\overline{BE}\end{align*}.

Step 6: Select the Compass tool to construct circles with radius equal to the length of \begin{align*}\overline{BE}\end{align*}.

• Press ENTER on \begin{align*}\overline{BE}\end{align*}. A dashed circle will appear and follow the pointer.
• Press ENTER on point \begin{align*}A\end{align*}. The compass circle is anchored at center \begin{align*}A\end{align*}.

Create a point of intersection of this circle with \begin{align*}\overline{AD}\end{align*}. Label this point \begin{align*}F\end{align*}.

Hide the compass circle.

Step 7: Use the Compass tool again to construct circles with centers at \begin{align*}C\end{align*} and \begin{align*}D\end{align*} and radius \begin{align*}= BE\end{align*}.

Create points of intersection of these circles with \begin{align*}\overline{DC}\end{align*} and \begin{align*}\overline{BC}\end{align*}. Label these points \begin{align*}G\end{align*} and \begin{align*}H\end{align*}.

Hide the compass circles.

Drag point \begin{align*}E\end{align*} to confirm that \begin{align*}F\end{align*}, \begin{align*}G\end{align*}, and \begin{align*}H\end{align*} all move as \begin{align*}E\end{align*} moves.

Step 8: Construct the quadrilateral \begin{align*}EFGH\end{align*}. Can you prove that this quadrilateral is a square?

Step 9: Use the Alph-Num tool to place the labels \begin{align*}x\end{align*}, \begin{align*}y\end{align*}, and \begin{align*}z\end{align*} on the figure as shown.

\begin{align*}\overline{BE}\end{align*} is labeled \begin{align*}x\end{align*}, therefore \begin{align*}AF = DG = CH = x\end{align*}.

\begin{align*}\overline{EA}\end{align*} is labeled \begin{align*}y\end{align*}, therefore \begin{align*}FD = GC = HB = y\end{align*}.

Since \begin{align*}ABCD\end{align*} is a square, each of the angles at \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, \begin{align*}C\end{align*} and \begin{align*}D\end{align*} are \begin{align*}90^\circ\end{align*}, so we have four congruent triangles, namely \begin{align*}\triangle{EFA}, \triangle{FGD}, \triangle{GHC}\end{align*} and \begin{align*}\triangle{HEB}\end{align*}.

Step 10: Start with this step if you are using the pre-constructed file “PYTHAG2”.

Let’s examine the algebra in this situation.

\begin{align*}ABCD\end{align*} is a square with sides of length \begin{align*}(x + y)\end{align*}.

The area of the square \begin{align*}ABCD\end{align*} is \begin{align*}(x + y)^2 = x^2 + 2xy + y^2\end{align*}.

Each of the triangles \begin{align*}\triangle{EFA}\end{align*}, \begin{align*}\triangle{FGD}\end{align*}, \begin{align*}\triangle{GHC}\end{align*} and \begin{align*}\triangle{HEB}\end{align*} is a right triangle with height \begin{align*}x\end{align*} and base \begin{align*}y\end{align*}. So, the area of each triangle is \begin{align*}\frac{1}{2}xy\end{align*}. The sum of the areas of the four triangles is \begin{align*}4 \cdot \frac{1}{2}xy=2xy\end{align*}.

\begin{align*}EFGH\end{align*} is a square with sides of length \begin{align*}z\end{align*}. So the area of \begin{align*}EFGH\end{align*} is \begin{align*}z^2\end{align*}.

Looking at the areas in the diagram we can conclude that:

\begin{align*}ABCD = \triangle{EFA} + \triangle{FGD} + \triangle{GHC} + \triangle{HEB} + EFGH\end{align*}

On the worksheet, substitute the area expressions (with variables \begin{align*}x\end{align*}, \begin{align*}y\end{align*}, and \begin{align*}z\end{align*}) into the equation above and simplify.

Step 11: Let’s look at this numerically as well to confirm what we just proved algebraically. Measure \begin{align*}\overline{BE}\end{align*}, \begin{align*}\overline{HB}\end{align*} and \begin{align*}\overline{EH}\end{align*}.

Note: Measure \begin{align*}\overline{HB}\end{align*} and \begin{align*}\overline{EH}\end{align*} by pressing ENTER on each endpoint, since these do not have separate segments constructed.

Use the Calculate tool to find the squares of these lengths.

Record your observations in the table for Question 6 on the worksheet.

Step 12: Find the sum of the squares of the lengths of segments \begin{align*}\overline{BE}\end{align*} and \begin{align*}\overline{HB}\end{align*}.

In the right triangle \begin{align*}HBE\end{align*}, is \begin{align*}BE^2 + HB^2 = EH^2\end{align*}?

Drag point \begin{align*}E\end{align*} to ensure that the relationship holds for other locations of the points \begin{align*}E\end{align*}, \begin{align*}F\end{align*}, \begin{align*}G\end{align*}, and \begin{align*}H\end{align*}.

What would happen if you dragged one of points \begin{align*}A\end{align*} or \begin{align*}B\end{align*}? Would the relationship still hold?

Answers Questions 7 and 8 on the worksheet.

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