Coordinate Proofs Math Example 3

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Example 3

medium
Use a coordinate proof to show that the segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side (Midsegment Theorem). Use triangle with vertices A(0,0)A(0,0), B(2a,0)B(2a,0), C(2b,2c)C(2b,2c).

Solution

  1. 1
    Step 1: Find the midpoints of ABAB and ACAC: MAB=(a,0)M_{AB} = (a, 0) and MAC=(b,c)M_{AC} = (b, c).
  2. 2
    Step 2: Find the slope of midsegment MABMACM_{AB}M_{AC}: slope=c−0b−a\text{slope} = \frac{c - 0}{b - a}. Find the slope of BCBC: slope of BC=2c−02b−2a=cb−a\text{slope of } BC = \frac{2c - 0}{2b - 2a} = \frac{c}{b-a}. The slopes are equal, so the segments are parallel.
  3. 3
    Step 3: Find the length of the midsegment: ∣MABMAC∣=(b−a)2+c2|M_{AB}M_{AC}| = \sqrt{(b-a)^2 + c^2}. Find the length of BCBC: ∣BC∣=(2b−2a)2+(2c)2=2(b−a)2+c2|BC| = \sqrt{(2b-2a)^2 + (2c)^2} = 2\sqrt{(b-a)^2+c^2}. The midsegment is exactly half of BCBC.

Answer

The midsegment is parallel to BCBC and has length 12∣BC∣\frac{1}{2}|BC|.
Choosing coordinates with even-numbered expressions (like 2a, 2b, 2c) avoids fractions when computing midpoints. Comparing slopes confirms parallelism, and comparing distances confirms the half-length property.

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 4 easy

Prove using coordinates that the diagonals of a square are perpendicular. Place the square with vert

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