Coordinate Proofs Math Example 4

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Example 4

easy
Prove using coordinates that the diagonals of a square are perpendicular. Place the square with vertices at (0,0)(0,0), (a,0)(a,0), (a,a)(a,a), (0,a)(0,a).

Solution

  1. 1
    Step 1: Identify the diagonals: from (0,0)(0,0) to (a,a)(a,a) with slope m1=aa=1m_1 = \frac{a}{a} = 1, and from (a,0)(a,0) to (0,a)(0,a) with slope m2=a−00−a=−1m_2 = \frac{a-0}{0-a} = -1.
  2. 2
    Step 2: Check perpendicularity: m1×m2=1×(−1)=−1m_1 \times m_2 = 1 \times (-1) = -1. Since the product of slopes is −1-1, the diagonals are perpendicular. ■\blacksquare

Answer

The diagonals are perpendicular because their slopes multiply to −1-1.
Two lines are perpendicular if and only if the product of their slopes equals -1. The diagonals of the square have slopes 1 and -1 respectively, confirming they are perpendicular.

About Coordinate Proofs

A method of proving geometric properties by placing figures on a coordinate plane and using algebraic formulas (distance, midpoint, slope) to verify relationships.

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More Coordinate Proofs Examples

Example 1 medium

Use a coordinate proof to show that the diagonals of a rectangle are equal in length. Place the rect

Example 2 hard

Use a coordinate proof to show that the midpoints of the sides of any quadrilateral form a parallelo

Example 3 medium

Use a coordinate proof to show that the segment connecting the midpoints of two sides of a triangle

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