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: Recall Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step2: Calculate \(XY\)

For \(X(2, -1)\) and \(Y(1, -7)\):
\(XY = \sqrt{(1 - 2)^2 + (-7 - (-1))^2} = \sqrt{(-1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37}\)

Step3: Calculate \(YZ\)

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

Step4: Calculate \(XZ\)

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

Step5: Calculate \(JK\)

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, no, wait: 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. Wait, \(J(5,2)\), \(K(6,6)\): \(x_2 - x_1 = 6 - 5 = 1\), \(y_2 - y_1 = 6 - 2 = 4\), so \(JK = \sqrt{1^2 + 4^2} = \sqrt{17}\)? Wait, but let's check again. Wait, maybe I made a mistake. Wait, no, let's proceed.

Step6: Calculate \(KL\)

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

Step7: Calculate \(JL\)

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, wait: \(J(5,2)\), \(L(1,7)\): \(x\) difference \(1 - 5 = -4\), \(y\) difference \(7 - 2 = 5\), so \(\sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}\)? Wait, but earlier \(XZ\) was \(\sqrt{65}\). Wait, maybe I messed up \(L\)'s coordinates. Wait, the problem says \(L(1, 7)\)? Wait, original problem: \(L(1,7)\)? Wait, the graph: Wait, maybe I misread \(L\)'s coordinates. Wait, the problem states \(L(1,7)\)? Wait, let's recheck the problem statement: "△JKL has vertices J(5, 2), K(6, 6), and L(1, 7)". Yes. So \(JL\): \(x\) from 5 to 1: -4, \(y\) from 2 to 7: +5. So \(\sqrt{(-4)^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}\). Wait, but \(XZ\) was \(\sqrt{65}\). Wait, maybe I made a mistake in \(XZ\). Wait, \(X(2,-1)\), \(Z(6,-8)\): \(x\) difference 4, \(y\) difference -7 (since -8 - (-1) = -7). So \(4^2 + (-7)^2 = 16 + 49 = 65\), so \(XZ = \sqrt{65}\). Correct.

Wait, but let's recalculate \(JK\) again: \(J(5,2)\), \(K(6,6)\): \(x\) difference 1, \(y\) difference 4. So \(1 + 16 = 17\), so \(JK = \sqrt{17}\). \(KL\): \(K(6,6)\), \(L(1,7)\): \(x\) difference -5, \(y\) difference 1. So \(25 + 1 = 26\), so \(KL = \sqrt{26}\). \(JL\): \(J(5,2)\), \(L(1,7)\): \(x\) difference -4, \(y\) difference 5. So \(16 + 25 = 41\), so \(JL = \sqrt{41}\).

Wait, but the problem has \(XY\), \(YZ\), \(XZ\) for △XYZ and \(JK\), \(KL\), \(JL\) for △JKL. Wait, maybe I misassigned the sides. Wait, the problem says "Find the side lengths below, then determine whether △XYZ ≅ △JKL." So we need to check if the corresponding sides are equal. Let's list all sides:

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

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

Wait, \(YZ = KL = \sqrt{26}\), but the other sides are not equal. Wait, maybe I misread the coordinates. Wait, let's check the graph again (from the image): Maybe \(L\) is (1,8)? Wait, the graph shows \(L\) at (1,8)? Wait, the y-axis on the graph: Let's see, the graph has y-axis with 8 at the top. So \(L\) is at (1,8)? Maybe the problem statement has a typo, or I misread \(L\)'s coordinates. Wait, the user's image: Let's look at the graph: \(L\) is at (1,8), \(J\) at (5,2), \(K\) at (6,6). So \(L(1,8)\), not (1,7). Ah, that's the mistake! So \(L(1,8)\). Let's recalculate with \(L(1,8)\).

Step5 (corrected): Calculate \(JK\) (J(5,2), K(6,6))

\(JK = \sqrt{(6 - 5)^2 + (6 - 2)^2} = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}\) (same as before)

Step6 (corrected): Calculate \(KL\) (K(6,6), L(1,8))

\(KL = \sqrt{(1 - 6)^2 + (8 - 6)^2} = \sqrt{(-5)^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29}\)? No, wait, (8 - 6) is 2, so \(25 + 4 = 29\)? No, that can't be. Wait, the graph: Let's count the grid. \(L\) is at (1,8), \(K\) at (6,6). So horizontal distance: 6 - 1 = 5 (wait, no, x-coordinate: \(L\) is at x=1, \(K\) at x=6, so 6 - 1 = 5 (horizontal difference 5), vertical difference: 6 - 8 = -2 (so 2 units down). So \(KL = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}\). No, that's not matching. Wait, maybe the graph has \(L\) at (1,8), \(J\) at (5,2), \(K\) at (6,6). Let's calculate \(JL\): \(J(5,2)\), \(L(1,8)\): horizontal difference 5 - 1 = 4 (wait, x=5 to x=1: -4), vertical difference 8 - 2 = 6. So \(JL = \sqrt{(-4)^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}\). No, this is confusing. Wait, let's use the graph's coordinates as per the image:

Looking at the graph:
- \(X\) is at (2, -1) (since x=2, y=-1)
- \(Y\) is at (1, -7) (x=1, y=-7)
- \(Z\) is at (6, -8) (x=6, y=-8)
- \(J\) is at (5, 2) (x=5, y=2)
- \(K\) is at (6, 6) (x=6, y=6)
- \(L\) is at (1, 8) (x=1, y=8)

Now recalculate:

For △XYZ:

- \(XY\): X(2,-1) to Y(1,-7): \(\sqrt{(1-2)^2 + (-7 - (-1))^2} = \sqrt{(-1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37}\)
- \(YZ\): Y(1,-7) to Z(6,-8): \(\sqrt{(6-1)^2 + (-8 - (-7))^2} = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26}\)
- \(XZ\): X(2,-1) to Z(6,-8): \(\sqrt{(6-2)^2 + (-8 - (-1))^2} = \sqrt{4^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65}\)

For △JKL:

- \(JK\): J(5,2) to K(6,6): \(\sqrt{(6-5)^2 + (6-2)^2} = \sqrt{1 + 16} = \sqrt{17}\) (no, wait, (6-2) is 4, correct. But let's check \(JL\): J(5,2) to L(1,8): \(\sqrt{(1-5)^2 + (8-2)^2} = \sqrt{(-4)^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}\). No, that's not matching. Wait, maybe the problem has a different set of coordinates. Alternatively, maybe the sides are \(XY\), \(YZ\), \(XZ\) and \(JL\), \(KL\), \(JK\) with correct correspondence.

But the initial mistake was \(L\)'s y-coordinate. Let's assume the problem statement had a typo, and \(L(1,8)\). Then:

- \(JL\) (J(5,2), L(1,8)): \(\sqrt{(1-5)^2 + (8-2)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}\) (no)
- \(KL\) (K(6,6), L(1,8)): \(\sqrt{(1-6)^2 + (8-6)^2} = \sqrt{25 + 4} = \sqrt{29}\) (no)
- \(JK\) (J(5,2), K(6,6)): \(\sqrt{1 + 16} = \sqrt{17}\) (no)

This is getting confusing. Let's go back to the original problem statement as given: "△JKL has vertices J(5, 2), K(6, 6), and L(1, 7)". So \(L(1,7)\). Then:

- \(JL\): J(5,2) to L(1,7): \(\sqrt{(1-5)^2 + (7-2)^2} = \sqrt{16 + 25} = \sqrt{41}\)
- \(KL\): K(6,6) to L(1,7): \(\sqrt{(1-6)^2 + (7-6)^2} = \sqrt{25 + 1} = \sqrt{26}\)
- \(JK\): J(5,2) to K(6,6): \(\sqrt{1 + 16} = \sqrt{17}\)

And △XYZ:

- \(XY\): X(2,-1) to Y(1,-7): \(\sqrt{1 + 36} = \sqrt{37}\)
- \(YZ\): Y(1,-7) to Z(6,-8): \(\sqrt{25 + 1} = \sqrt{26}\)
- \(XZ\): X(2,-1) to Z(6,-8): \(\sqrt{16 + 49} = \sqrt{65}\)

So \(YZ = KL = \sqrt{26}\), but the other sides are not equal. So the side lengths:

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

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

But the problem asks for the side lengths, so we'll provide the exact values as calculated.

Answer:

\(XY = \sqrt{37}\), \(YZ = \sqrt{26}\), \(XZ = \sqrt{65}\), \(JK = \sqrt{17}\), \(KL = \sqrt{26}\), \(JL = \sqrt{41}\)

(Note: If the graph shows \(L(1,8)\), the calculations would change, but based on the problem statement, we used \(L(1,7)\). The key is applying the distance formula correctly.)