Question

note that △xyz has vertices x(2, −1), y(1, −7), and z(6, −8) and △jkl has vertices j(5, 2), k(6, 6), and l(1, 7). find the side lengths below, then determine whether △xyz ≅ △jkl. give exact answers (not decimal approximations).
part 1 of 2
xy =
yz =
xz =
jk =
kl =
jl =

Explanation:

Step1: Find \( XY \)

Use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). For \( X(2, -1) \) and \( Y(1, -7) \):
\( XY = \sqrt{(1 - 2)^2 + (-7 - (-1))^2} = \sqrt{(-1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37} \)

Step2: Find \( YZ \)

For \( Y(1, -7) \) and \( Z(6, -8) \):
\( YZ = \sqrt{(6 - 1)^2 + (-8 - (-7))^2} = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \)

Step3: Find \( XZ \)

For \( X(2, -1) \) and \( Z(6, -8) \):
\( XZ = \sqrt{(6 - 2)^2 + (-8 - (-1))^2} = \sqrt{4^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65} \)

Step4: Find \( JK \)

For \( J(5, 2) \) and \( K(6, 6) \):
\( JK = \sqrt{(6 - 5)^2 + (6 - 2)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \) (Wait, correction: Wait, \( J(5,2) \), \( K(6,6) \): \( x \) difference 1, \( y \) difference 4. So \( \sqrt{1 + 16} = \sqrt{17} \)? Wait no, original problem: Wait, maybe I misread \( K \). Wait, \( K(6,6) \)? Wait, the problem says \( K(6,6) \)? Wait, no, looking back: \( J(5,2) \), \( K(6,6) \), \( L(1,7) \). Wait, no, maybe I made a mistake. Wait, let's recheck \( JK \):

Wait, \( J(5,2) \), \( K(6,6) \): \( \Delta x = 6 - 5 = 1 \), \( \Delta y = 6 - 2 = 4 \). So \( JK = \sqrt{1^2 + 4^2} = \sqrt{17} \). Wait, but maybe the graph has different coordinates? Wait, no, the given coordinates: \( J(5,2) \), \( K(6,6) \), \( L(1,7) \).

Step5: Find \( KL \)

For \( K(6,6) \) and \( L(1,7) \):
\( KL = \sqrt{(1 - 6)^2 + (7 - 6)^2} = \sqrt{(-5)^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \)

Step6: Find \( JL \)

For \( J(5,2) \) and \( L(1,7) \):
\( JL = \sqrt{(1 - 5)^2 + (7 - 2)^2} = \sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \)? Wait, no, this is conflicting. Wait, maybe I misread the coordinates. Wait, the problem says \( \triangle XYZ \) with \( X(2,-1) \), \( Y(1,-7) \), \( Z(6,-8) \); \( \triangle JKL \) with \( J(5,2) \), \( K(6,6) \), \( L(1,7) \).

Wait, let's redo \( JK \): \( J(5,2) \), \( K(6,6) \): \( x \) from 5 to 6 (1), \( y \) from 2 to 6 (4). So \( JK = \sqrt{1 + 16} = \sqrt{17} \). \( KL \): \( K(6,6) \) to \( L(1,7) \): \( x \) 6 to 1 (-5), \( y \) 6 to 7 (1). So \( \sqrt{25 + 1} = \sqrt{26} \). \( JL \): \( J(5,2) \) to \( L(1,7) \): \( x \) 5 to 1 (-4), \( y \) 2 to 7 (5). So \( \sqrt{16 + 25} = \sqrt{41} \). Wait, but earlier \( XZ \) was \( \sqrt{65} \), \( XY \sqrt{37} \), \( YZ \sqrt{26} \). \( KL \) is \( \sqrt{26} \), so \( YZ = KL = \sqrt{26} \). \( XY = \sqrt{37} \), \( JK = \sqrt{17} \) (not equal). \( XZ = \sqrt{65} \), \( JL = \sqrt{41} \) (not equal). So the side lengths:

\( XY = \sqrt{37} \), \( YZ = \sqrt{26} \), \( XZ = \sqrt{65} \)

\( JK = \sqrt{17} \), \( KL = \sqrt{26} \), \( JL = \sqrt{41} \)

(Note: Maybe I misread \( K \)'s coordinates. Wait, the graph: Let's check the graph. If \( J \) is at (5, -5)? Wait, the user's graph: Maybe the coordinates were misread. Wait, the original problem's graph: Let's re-express. Maybe \( J(5, -5) \), \( K(6, -6) \), \( L(1, 7) \)? No, the text says \( J(5,2) \), \( K(6,6) \), \( L(1,7) \). So proceeding with given coordinates.)

Answer:

\( XY = \sqrt{37} \), \( YZ = \sqrt{26} \), \( XZ = \sqrt{65} \), \( JK = \sqrt{17} \), \( KL = \sqrt{26} \), \( JL = \sqrt{41} \) (For the first part, the side lengths of \( \triangle XYZ \) are \( XY = \sqrt{37} \), \( YZ = \sqrt{26} \), \( XZ = \sqrt{65} \); for \( \triangle JKL \), \( JK = \sqrt{17} \), \( KL = \sqrt{26} \), \( JL = \sqrt{41} \))