Geometry
You’re going to construct a triangle and a quadrilateral with the following specifications:
- 1.
- is °, the line is long, and is °. Construct the triangle .
What kind of triangle is this? How large is ?
- 2.
- The area of the triangle in Exercise 1 is half the area of the rectangle .
How long are the sides in the rectangle? How long is the diagonal?
Example 1
Make an auxiliary figure, which is a small figure of what you’re going to construct. Mark the angles, lengths, and any other information.
Begin the construction:
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Draw a long line.
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Mark point and measure along the line to point .
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Construct a ° angle at .
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Construct a ° angle at .
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Name the intersection between and , .
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This is a right triangle.
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.
Example 2
Continue the construction by constructing the quadrilateral :
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Construct a normal on the line at .
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Construct a normal on at . Name the intersection between the two normals . You have now constructed the rectangle .
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Because the rectangle is made up of two “°-°-°” triangles, the hypotenuse is twice the length of the shortest leg. That means the diagonal is .
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To find the height of the triangle, you use the Pythagorean theorem:
You’ve found that the sides are and , and the diagonal is .