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Geometer’s Sketchpad – Pythagoras and Special Right Triangles

These instructions should be returned to Mr. Powers!

Today you and a partner will CONSTRUCT a right triangle, the squares of its sides, and calculate the areas of the squares to show that the Pythagorean Theorem is true for all right triangles. You will also construct the special right triangles and calculate their measurements to show their properties.

Reminder – a construction is different from a sketch. A construction is based on Intersections, and remains true if you drag/stretch the object in Geometer’s Sketchpad.

Part 1: Pythagorean Theorem.

Draw a segment. Select one the endpoints and segment and go the “construct Perpendicular Line” under the Construct Tab. Draw a segment overlapping the perpendicular line and starting at the endpoint. Connect the perpendicular segments to make a triangle. Hide the perpendicular line.

Construct a square on each segment of the triangle by the following process. Construct a circle with a center on one point of the triangle and goes to another vertex of the triangle. This is a circle with a radius of the length of the side of the triangle. Construct perpendicular lines through the vertices and use the intersection with the circles to get the length of the square. Hide the parts you aren’t using and repeat until you have a square constructed on each side of the right triangle.

Find the area of each square by selecting it and using the Measure tab. Before you can directly measure the area however, you must construct the interior of each polygon. Select the vertices of one square and use the construct tab to construct the Quadrilateral Interior. Now select the colored shape and use the Measure tab to find the area. You can also find the area using the side lengths and the formula for the area of a square.

Use the Calculate option under the Measure tab to show that the smaller areas add up to the larger area.

Part 2: Special Right Triangles

On two new pages, Construct the two special right triangles, measure their angles, side lengths, and calculate their side lengths based on the formulas we learned to show that the measurement is the same as the calculation.

To construct a 45-45-90 triangle construct a square (like you did in Part 1) and construct a diagonal. Hide what you don’t need.

To construct a 30-60-90 triangle cut an equilateral triangle in half. Construct the equilateral triangle that we learned last semester by overlapping two circles of the same radius and connecting centers and the intersections. Then construct the line through the triangle.

Bonus!

When you finish and have the above checked by me you can try these problems. (Use your book!)

1. Construct a regular Hexagon.

2. Construct a regular Pentagon.

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