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Prove that the diagonals of a rhombus (parallelogram with 4 equal sides) are perpendicular to each other.

Prove that the diagonals of a rhombus (parallelogram with 4 equal sides) are perpendicular to each other.

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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**Prove that the diagonals of a rhombus (parallelogram with 4 equal sides) are perpendicular to each other.** To prove this, consider a rhombus \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(O\). ### Steps of the Proof: 1. **Properties of a Rhombus:** - All sides are equal, i.e., \(AB = BC = CD = DA\). - Opposite angles are equal, and adjacent angles are supplementary. 2. **Diagonals of a Rhombus:** - Diagonals bisect each other at \(O\). 3. **Apply the Pythagorean Theorem:** - Consider triangles \(AOB\), \(BOC\), \(COD\), and \(DOA\). - Since all sides of the rhombus are equal, \(AB = BC = CD = DA\). 4. **Right Triangle Condition:** - For triangles around point \(O\), \(OA = OB = OC = OD\). - By symmetry and the properties of diagonals in a rhombus, each pair of triangles is congruent. - The combinations of sides form right angles due to the equality of the lengths and the geometric placement. 5. **Conclusion:** - The diagonals of a rhombus intersect at 90 degrees, proving they are perpendicular to each other. ### Diagram Description: A rhombus should be illustrated with diagonals \(AC\) and \(BD\) intersecting at point \(O\). Each side should be marked as equal, and the intersection at \(O\) should be highlighted to show perpendicularity with a small square indicating the right angle. This completes the proof that the diagonals of a rhombus are perpendicular to one another.
Transcribed Image Text:**Prove that the diagonals of a rhombus (parallelogram with 4 equal sides) are perpendicular to each other.** To prove this, consider a rhombus \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(O\). ### Steps of the Proof: 1. **Properties of a Rhombus:** - All sides are equal, i.e., \(AB = BC = CD = DA\). - Opposite angles are equal, and adjacent angles are supplementary. 2. **Diagonals of a Rhombus:** - Diagonals bisect each other at \(O\). 3. **Apply the Pythagorean Theorem:** - Consider triangles \(AOB\), \(BOC\), \(COD\), and \(DOA\). - Since all sides of the rhombus are equal, \(AB = BC = CD = DA\). 4. **Right Triangle Condition:** - For triangles around point \(O\), \(OA = OB = OC = OD\). - By symmetry and the properties of diagonals in a rhombus, each pair of triangles is congruent. - The combinations of sides form right angles due to the equality of the lengths and the geometric placement. 5. **Conclusion:** - The diagonals of a rhombus intersect at 90 degrees, proving they are perpendicular to each other. ### Diagram Description: A rhombus should be illustrated with diagonals \(AC\) and \(BD\) intersecting at point \(O\). Each side should be marked as equal, and the intersection at \(O\) should be highlighted to show perpendicularity with a small square indicating the right angle. This completes the proof that the diagonals of a rhombus are perpendicular to one another.
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