When two triangles are placed on a coordinate grid, we can determine whether or not they are congruent using the side-side-side (SSS) test for congruence, by computing the length of each side. The most important formula, in this case, is the formula for the distance between two points.

The distance $d$d between the points $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2) on the $\left(x,y\right)$(x,y)-plane is given by

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$d=√(x2−x1)2+(y2−y1)2.

In words, it is the square root of the sum of the squares of the differences between both the $x$x-values and the $y$y-values.

This formula will allow us to directly compute the length of each side of two different triangles, and if all three sides have a matching side on the other triangle with the same length, we conclude they are congruent by SSS.

Show that $\triangle RST$△RST is congruent to $\triangle CDE$△CDE.

Think: We will use the distance formula to compute the lengths of all six segments forming the sides of each triangle. A good intermediate step would be to write down the coordinates of each point.

$R:\left(-5,4\right)\text{, }S:\left(-2,8\right)\text{, }T:\left(0,-8\right)\text{, }C:\left(8,-3\right)\text{, }D:\left(5,-7\right)\text{, }E:\left(3,9\right)$R:(−5,4), S:(−2,8), T:(0,−8), C:(8,−3), D:(5,−7), E:(3,9)

Do: Applying the formula, we find,

$RS$RS | $=$= | $\sqrt{\left(\left(-2\right)-\left(-5\right)\right)^2+\left(8-4\right)^2}$√((−2)−(−5))2+(8−4)2 | $=$= | $\sqrt{3^2+4^2}$√32+42 | $=$= | $5$5 |

$ST$ST | $=$= | $\sqrt{\left(0-\left(-2\right)\right)^2+\left(\left(-8\right)-8\right)^2}$√(0−(−2))2+((−8)−8)2 | $=$= | $\sqrt{2^2+\left(-16\right)^2}$√22+(−16)2 | $=$= | $\sqrt{260}$√260 |

$TR$TR | $=$= | $\sqrt{\left(\left(-5\right)-0\right)^2+\left(4-\left(-8\right)\right)^2}$√((−5)−0)2+(4−(−8))2 | $=$= | $\sqrt{\left(-5\right)^2+12^2}$√(−5)2+122 | $=$= | $13$13 |

$CD$CD | $=$= | $\sqrt{\left(5-8\right)^2+\left(\left(-7\right)-\left(-3\right)\right)^2}$√(5−8)2+((−7)−(−3))2 | $=$= | $\sqrt{\left(-3\right)^2+\left(-4\right)^2}$√(−3)2+(−4)2 | $=$= | $5$5 |

$DE$DE | $=$= | $\sqrt{\left(3-5\right)^2+\left(9-\left(-7\right)\right)^2}$√(3−5)2+(9−(−7))2 | $=$= | $\sqrt{\left(-2\right)^2+16^2}$√(−2)2+162 | $=$= | $\sqrt{260}$√260 |

$EC$EC | $=$= | $\sqrt{\left(8-3\right)^2+\left(\left(-3\right)-9\right)^2}$√(8−3)2+((−3)−9)2 | $=$= | $\sqrt{5^2+\left(-12\right)^2}$√52+(−12)2 | $=$= | $13$13 |

Notice that we have been very careful of the minus signs involved – this is a cause of a lot of errors in these sorts of calculations. Since segments of equal length are congruent, we can conclude that

$\overline{RS}\cong\overline{CD}\text{, }\overline{ST}\cong\overline{DE}\text{, }\overline{TR}\cong\overline{EC}$RS≅CD, ST≅DE, TR≅EC

so $\triangle RST\cong\triangle CDE$△RST≅△CDE by SSS.

Determine whether or not $\triangle WXY$△WXY and $\triangle YZW$△YZW are congruent.

Think: It seems as though $\overline{WX}$WX and $\overline{YZ}$YZ are different lengths, and all we have to do to disprove congruence is to show that the length of the side of one triangle doesn’t match the lengths of any sides of the other triangle. It is still a good idea to write down the coordinates of each point before we begin.

$W:\left(-4,9\right)\text{, }X:\left(6,10\right)\text{, }Y:\left(6,-7\right)\text{, }Z:\left(-5,-8\right)$W:(−4,9), X:(6,10), Y:(6,−7), Z:(−5,−8)

Do: Calculating the lengths of the sides of $\triangle WXY$△WXY,

$WX$WX | $=$= | $\sqrt{\left(6-\left(-4\right)\right)^2+\left(10-9\right)^2}$√(6−(−4))2+(10−9)2 | $=$= | $\sqrt{10^2+1^2}$√102+12 | $=$= | $\sqrt{101}$√101 |

$XY$XY | $=$= | $\sqrt{\left(6-6\right)^2+\left(\left(-7\right)-10\right)^2}$√(6−6)2+((−7)−10)2 | $=$= | $\sqrt{0^2+\left(-17\right)^2}$√02+(−17)2 | $=$= | $17$17 |

$YW$YW | $=$= | $\sqrt{\left(\left(-4\right)-6\right)^2+\left(9-\left(-7\right)\right)^2}$√((−4)−6)2+(9−(−7))2 | $=$= | $\sqrt{\left(-10\right)^2+16^2}$√(−10)2+162 | $=$= | $\sqrt{356}$√356 |

Similarly, we calculate the length of $\overline{YZ}$YZ.

$YZ$YZ | $=$= | $\sqrt{\left(\left(-5\right)-6\right)^2+\left(\left(-8\right)-\left(-7\right)\right)^2}$√((−5)−6)2+((−8)−(−7))2 | $=$= | $\sqrt{\left(-11\right)^2+\left(-1\right)^2}$√(−11)2+(−1)2 | $=$= | $\sqrt{122}$√122 |

Since the length of $\overline{YZ}$YZ does not match the lengths of any side of $\triangle WXY$△WXY, we conclude that the two triangles are not congruent.

Reflect: How else might we prove that two triangles on the coordinate plane are congruent?

We wish to determine if the pair of triangles on the coordinate plane below are congruent.

What translations move the point $X$X to the point $B$B?

Translate $\editable{}$ units left and $\editable{}$ units down.

Apply the translation from part (a) to the other two points of the triangle $\triangle YXZ$△YXZ.

Which two options represent the results?

$Z$Z translates to C

$Y$Y translates to A

$Z$Z translates to A

$Y$Y translates to C

$Z$Z does not translate to any vertex of $\triangle ABC$△ABC.

$Y$Y does not translate to any vertex of $\triangle ABC$△ABC.

Are $\triangle ABC$△ABC and $\triangle YXZ$△YXZ congruent?

Yes

No

Consider the two triangles drawn in the diagram below.

Are the triangles $\triangle YBC$△YBC and $\triangle YXZ$△YXZ congruent?

Yes

No

We wish to determine if the pair of triangles on the coordinate plane below are congruent.

First, find the exact length of $\overline{AB}$AB.

Using the same method as the previous part, or otherwise, complete the table of segment lengths.

Triangle | $\triangle ABC$△ABC | $\triangle XYZ$△XYZ | ||||

Segment | $\overline{AB}$AB | $\overline{BC}$BC | $\overline{CA}$CA | $\overline{XY}$XY | $\overline{YZ}$YZ | $\overline{ZX}$ZX |

Length | $\sqrt{29}$√29 | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |

Are $\triangle ABC$△ABC and $\triangle ZYX$△ZYX congruent?

Yes

No

Use coordinates to prove simple geometric theorems algebraically, including the distance formula and its relationship to the Pythagorean Theorem