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All courses Math I · G-GPE.4 44 of 59
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.

Use the distance formula to prove segments congruent

Problem
Use the distance formula to calculate the lengths of segments A(0,0)B(3,4) and C(1,1)D(4,5), then state the congruence conclusion.
Source coordinate plane: Segment AB runs from A(0,0) to B(3,4), and segment CD runs from C(1,1) to D(4,5); no lengths or congruence conclusion are shown. Open full size
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Hint

Use the distance formula on each segment by subtracting the x-coordinates and y-coordinates first.

If two segments have the same length, then the segments are congruent.

Solution walkthrough

01

Set up the distance formula

\[\text{distance}~=~\text{sqrt}((x2-x1)^2+(y2-y1)^2)\]

Segment length comes from the horizontal and vertical coordinate differences between its endpoints.

02

Calculate AB

\[AB=\text{sqrt}((3-0)^2+(4-0)^2)=\text{sqrt}(9+16)=\text{sqrt}(25)=5\]

The prompt gives A(0,0) and B(3,4), producing coordinate changes 3 and 4.

03

Calculate CD

\[CD=\text{sqrt}((4-1)^2+(5-1)^2)=\text{sqrt}(9+16)=\text{sqrt}(25)=5\]

The prompt gives C(1,1) and D(4,5), which also differ by 3 horizontally and 4 vertically.

04

Compare and conclude

\[AB~=~5~\text{and}~CD~=~5,~\text{so}~\text{the}~\text{segments}~\text{are}~\text{congruent}.\]

Equal segment lengths establish segment congruence.

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Another way

  1. Both segments have displacement (3,4), so each is the hypotenuse of a 3-by-4 right triangle and has length 5.

!

Common mistake

Do not stop at the squared distance 25 or add coordinate differences as 3+4. The distance formula requires the square root of the sum of squares.

Worked coordinate diagram: AB = 5 and CD = 5, so the segments are congruent.
conclusion: segments are congruent; length a: 5; length b: 5