Question

Given the coordinates, determine whether PQR & XYZ are congruent

P(5,-4), Q(-3, 7), R(0, 2), X(-2,-1), Y(9, 7), Z(3, 2)


PQ=

XY=

QR=

YZ=

PR=

XZ=


Are the triangles congruent? If yes, explain your reasoning and write a congruence statement.

Answer 1

The triangles are congruent

Explanation:

Given

`P(5,-4), Q(-3, 7), R(0, 2), X(-2,-1), Y(9, 7), Z(3, 2)`

Required

Determine if PQR = XYZ

To do this, we make use of distance formula.

`D = √((x_2-x_1)^2+(y_2-y_1)^2)`

For PQ:

`(x_1,y_1) = (5,-4)` and
`(x_2,y_2) = (-3,7)`

So:

`PQ = √((-3-5)^2+(7-(-4))^2)`

`PQ = √((-8)^2+(7+4)^2)`

`PQ = √((-8)^2+(11)^2)`

`PQ = √(64+121)`

`PQ = √(185)`

For XY:

`(x_1,y_1) = (-2,-1)`

`(x_2,y_2) = (9,7)`

So:

`XY = √((9-(-2))^2+(7-(-1))^2)`

`XY = √((9+2)^2+(7+1)^2)`

`XY = √((11)^2+(8)^2)`

`XY = √(121+64)`

`XY = √(185)`

For QR:

`(x_1,y_1) = (-3,7)` and
`(x_2,y_2) = (0,2)`

So:

`QR = √((0-(-3))^2 + (2-7)^2)`

`QR = √((0+3)^2 + (-5)^2)`

`QR = √((3)^2 + 25)`

`QR = √(9 + 25)`

`QR = √(34)`

For YZ:

`(x_1,y_1) = (9,7)` and
`(x_2,y_2) = (3,2)`

So:

`YZ = √((3-9)^2+(2-7)^2)`

`YZ = √((-6)^2+(-5)^2)`

`YZ = √(36+25)`

`YZ = √(61)`

For PR:

`(x_1,y_1) = (5,-4)` and
`(x_2,y_2) = (0,2)`

So:

`PR = √((0-5)^2+(2-(-4))^2)`

`PR = √((-5)^2+(2+4)^2)`

`PR = √((-5)^2+(6)^2)`

`PR = √(25+36)`

`PR = √(61)`

For XZ:

`(x_1,y_1) = (-2,-1)` and
`(x_2,y_2) = (3,2)`

So:

`XZ = √((3-(-2))^2+(2-(-1))^2)`

`XZ = √((3+2)^2+(2+1)^2)`

`XZ = √(5^2+3^2)`

`XZ = √(34)`

From the calculations above:

`PQ = XY = \sqrt{185`

`QR = XZ = \sqrt{34`

`PR = YZ = \sqrt{61`

Hence, the triangles are congruent base on SSS (Sides- Sides-Sides)