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Theorem. The diagonals of a rectangle bisect each other and have the same length.

Theorem. The diagonals of a rectangle bisect each other and have the same length.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Theorem.** *The diagonals of a rectangle bisect each other and have the same length.* **Proof.** Let \( \square ABCD \) be a rectangle, i.e., a quadrilateral that has insert defining characteristic. Draw diagonals \( \overline{AC} \) and \( \overline{BD} \). *Insert figure that illustrates the setup* Explain why the diagonals bisect each other. (Do not work too hard. Make use of a combination of theorems we have already proven.) Next we will show the diagonals have the same length. Note that \( \overline{AC} \) is a side of triangle blah and \( \overline{BD} \) is a side of triangle blah. Thus it suffices to show triangles blah and blah are congruent because then their corresponding sides will be congruent. *Insert figure highlighting the two triangles of interest* Write an argument for why the triangles you identified in the previous step are congruent. *Insert a diagram that uses color or line style to mark the segments or angles you are using to justify congruence of the triangles* Therefore the corresponding sides \( \overline{AC} \) and \( \overline{BD} \) are congruent. Thus the diagonals of a rectangle bisect each other and have the same length. □
Transcribed Image Text:**Theorem.** *The diagonals of a rectangle bisect each other and have the same length.* **Proof.** Let \( \square ABCD \) be a rectangle, i.e., a quadrilateral that has insert defining characteristic. Draw diagonals \( \overline{AC} \) and \( \overline{BD} \). *Insert figure that illustrates the setup* Explain why the diagonals bisect each other. (Do not work too hard. Make use of a combination of theorems we have already proven.) Next we will show the diagonals have the same length. Note that \( \overline{AC} \) is a side of triangle blah and \( \overline{BD} \) is a side of triangle blah. Thus it suffices to show triangles blah and blah are congruent because then their corresponding sides will be congruent. *Insert figure highlighting the two triangles of interest* Write an argument for why the triangles you identified in the previous step are congruent. *Insert a diagram that uses color or line style to mark the segments or angles you are using to justify congruence of the triangles* Therefore the corresponding sides \( \overline{AC} \) and \( \overline{BD} \) are congruent. Thus the diagonals of a rectangle bisect each other and have the same length. □
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