Question

lets try: δjkl has a perimeter of 24 units. determine which ordered pair could be the coordinates of point k. select all that apply. (0, 2) (3, -4) (-3, 4) (5, 4) (-5, 2) (3, 5)

Explanation:

First, we need to find the coordinates of points \( J \) and \( L \) from the graph. From the graph, point \( J \) is at \( (-5, -4) \) and point \( L \) is at \( (3, 2) \).

Step 1: Calculate the distance between \( J \) and \( L \)

Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) = (-5, -4) \) and \( (x_2, y_2) = (3, 2) \).

\[ \begin{align*} d_{JL} &= \sqrt{(3 - (-5))^2 + (2 - (-4))^2}\\ &= \sqrt{(8)^2 + (6)^2}\\ &= \sqrt{64 + 36}\\ &= \sqrt{100}\\ &= 10 \end{align*} \]

So the length of \( JL \) is 10 units. Since the perimeter of \( \triangle JKL \) is 24 units, the sum of the lengths of \( JK \) and \( KL \) must be \( 24 - 10 = 14 \) units.

Step 2: Check each ordered pair for point \( K \)

We will calculate the distance from \( J(-5, -4) \) to \( K \) and from \( L(3, 2) \) to \( K \), then check if their sum is 14.

For \( K(0, 2) \):

- Distance \( JK \): \( \sqrt{(0 - (-5))^2 + (2 - (-4))^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.81 \)
- Distance \( KL \): \( \sqrt{(0 - 3)^2 + (2 - 2)^2} = \sqrt{9 + 0} = 3 \)
- Sum: \( 7.81 + 3 \approx 10.81 eq 14 \) → Not valid.

For \( K(3, -4) \):

- Distance \( JK \): \( \sqrt{(3 - (-5))^2 + (-4 - (-4))^2} = \sqrt{64 + 0} = 8 \)
- Distance \( KL \): \( \sqrt{(3 - 3)^2 + (-4 - 2)^2} = \sqrt{0 + 36} = 6 \)
- Sum: \( 8 + 6 = 14 \) → Valid.

For \( K(-3, 4) \):

- Distance \( JK \): \( \sqrt{(-3 - (-5))^2 + (4 - (-4))^2} = \sqrt{4 + 64} = \sqrt{68} \approx 8.25 \)
- Distance \( KL \): \( \sqrt{(-3 - 3)^2 + (4 - 2)^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32 \)
- Sum: \( 8.25 + 6.32 \approx 14.57 eq 14 \) → Not valid (close but not exact, maybe miscalculation? Wait, let's recalculate. Wait, \( JK \): \( (-3 - (-5)) = 2 \), \( (4 - (-4)) = 8 \), so \( \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} \approx 8.246 \). \( KL \): \( (-3 - 3) = -6 \), \( (4 - 2) = 2 \), so \( \sqrt{(-6)^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.324 \). Sum is ~14.57, not 14. Wait, maybe I made a mistake. Wait, perimeter is 24, \( JL = 10 \), so \( JK + KL = 14 \). Let's check another point.

For \( K(5, 4) \):

- Distance \( JK \): \( \sqrt{(5 - (-5))^2 + (4 - (-4))^2} = \sqrt{100 + 64} = \sqrt{164} \approx 12.81 \)
- Distance \( KL \): \( \sqrt{(5 - 3)^2 + (4 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \)
- Sum: \( 12.81 + 2.83 \approx 15.64 eq 14 \) → Not valid.

For \( K(-5, 2) \):

- Distance \( JK \): \( \sqrt{(-5 - (-5))^2 + (2 - (-4))^2} = \sqrt{0 + 36} = 6 \)
- Distance \( KL \): \( \sqrt{(-5 - 3)^2 + (2 - 2)^2} = \sqrt{64 + 0} = 8 \)
- Sum: \( 6 + 8 = 14 \) → Valid.

For \( K(3, 5) \):

- Distance \( JK \): \( \sqrt{(3 - (-5))^2 + (5 - (-4))^2} = \sqrt{64 + 81} = \sqrt{145} \approx 12.04 \)
- Distance \( KL \): \( \sqrt{(3 - 3)^2 + (5 - 2)^2} = \sqrt{0 + 9} = 3 \)
- Sum: \( 12.04 + 3 = 15.04 eq 14 \) → Not valid.

Wait, earlier for \( K(3, -4) \), \( JK = 8 \), \( KL = 6 \), sum 14. For \( K(-5, 2) \), \( JK = 6 \), \( KL = 8 \), sum 14. Let's recheck \( K(-3, 4) \). Wait, maybe I made a mistake in coordinates. Wait, point \( J \) is at \( (-5, -4) \) (from the graph: x=-5, y=-4), point \( L \) is at \( (3, 2) \) (x=3, y=2). Let's check \( K(-3, 4) \) again:

\( JK \): \( x \)-difference: -3 - (-5) = 2, \( y \)-difference: 4 - (-4) = 8. So \( \sqrt{2^2 + 8^2} = \sqrt{4 + 64} = \sqrt{68} \approx 8.246 \)

\( KL \): \( x \)-difference: -3 - 3 = -6, \( y \)-difference: 4 - 2 = 2. So \( \sqrt{(-6)^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.324 \)

Sum: 8.246 + 6.324 ≈ 14.57, which is more than 14. So not valid.

Wait, maybe the coordinates of \( J \) and \( L \) are different. Let me re-examine the graph. The graph has \( J \) at (-5, -4) (since it's at x=-5, y=-4) and \( L \) at (3, 2) (x=3, y=2). Let's check \( K(0,2) \) again:

\( JK \): \( 0 - (-5) = 5 \), \( 2 - (-4) = 6 \), so \( \sqrt{25 + 36} = \sqrt{61} \approx 7.81 \)

\( KL \): \( 0 - 3 = -3 \), \( 2 - 2 = 0 \), so \( \sqrt{9 + 0} = 3 \). Sum ≈ 10.81, not 14.

Wait, maybe I made a mistake in the perimeter. The perimeter is 24, so \( JK + KL + JL = 24 \). We found \( JL = 10 \), so \( JK + KL = 14 \). Let's check \( K(3, -4) \):

\( JK \): distance from (-5, -4) to (3, -4): since y-coordinates are the same, it's \( |3 - (-5)| = 8 \).

\( KL \): distance from (3, -4) to (3, 2): x-coordinates same, so \( |2 - (-4)| = 6 \).

So \( 8 + 6 + 10 = 24 \). Correct.

\( K(-5, 2) \):

\( JK \): distance from (-5, -4) to (-5, 2): x-coordinates same, so \( |2 - (-4)| = 6 \).

\( KL \): distance from (-5, 2) to (3, 2): y-coordinates same, so \( |3 - (-5)| = 8 \).

So \( 6 + 8 + 10 = 24 \). Correct.

Wait, what about \( K(-3, 4) \):

\( JK \): \( \sqrt{(-3 + 5)^2 + (4 + 4)^2} = \sqrt{4 + 64} = \sqrt{68} \approx 8.246 \)

\( KL \): \( \sqrt{(-3 - 3)^2 + (4 - 2)^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.324 \)

Sum: ~14.57, so \( 14.57 + 10 = 24.57 eq 24 \). So not valid.

\( K(5,4) \):

\( JK \): \( \sqrt{(5 + 5)^2 + (4 + 4)^2} = \sqrt{100 + 64} = \sqrt{164} \approx 12.81 \)

\( KL \): \( \sqrt{(5 - 3)^2 + (4 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \)

Sum: ~15.64, \( 15.64 + 10 = 25.64 eq 24 \).

\( K(3,5) \):

\( JK \): \( \sqrt{(3 + 5)^2 + (5 + 4)^2} = \sqrt{64 + 81} = \sqrt{145} \approx 12.04 \)

\( KL \): \( \sqrt{(3 - 3)^2 + (5 - 2)^2} = \sqrt{0 + 9} = 3 \)

Sum: ~15.04, \( 15.04 + 10 = 25.04 eq 24 \).

\( K(0,2) \):

\( JK \): \( \sqrt{(0 + 5)^2 + (2 + 4)^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.81 \)

\( KL \): \( \sqrt{(0 - 3)^2 + (2 - 2)^2} = \sqrt{9 + 0} = 3 \)

Sum: ~10.81, \( 10.81 + 10 = 20.81 eq 24 \).

So the valid points are \( (3, -4) \) and \( (-5, 2) \). Wait, but let's check again. Wait, maybe the coordinates of \( J \) are (-5, -4) and \( L \) are (3, 2). Let's confirm the distance \( JL \):

\( x \)-difference: 3 - (-5) = 8, \( y \)-difference: 2 - (-4) = 6. So \( \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \). Correct. So perimeter is 24, so \( JK + KL = 14 \).

For \( K(3, -4) \):

\( JK \): distance from (-5, -4) to (3, -4) is 8 (horizontal distance), \( KL \): distance from (3, -4) to (3, 2) is 6 (vertical distance). So 8 + 6 = 14. Correct.

For \( K(-5, 2) \):

\( JK \): distance from (-5, -4) to (-5, 2) is 6 (vertical distance), \( KL \): distance from (-5, 2) to (3, 2) is 8 (horizontal distance). So 6 + 8 = 14. Correct.

Any other points? Let's check \( K(-3, 4) \) again. Wait, maybe I miscalculated. \( JK \): (-3,4) to (-5,-4): \( x \)-diff: 2, \( y \)-diff: 8, so \( \sqrt{4 + 64} = \sqrt{68} \approx 8.246 \). \( KL \): (-3,4) to (3,2): \( x \)-diff: -6, \( y \)-diff: 2, so \( \sqrt{36 + 4} = \sqrt{40} \approx 6.324 \). Sum: ~14.57, which is more than 14, so \( 14.57 + 10 = 24.57 \), which is more than 24. So not valid.

So the valid ordered pairs are \( (3, -4) \) and \( (-5, 2) \).

Answer:

The ordered pairs that could be the coordinates of point \( K \) are \( \boldsymbol{(3, -4)} \) and \( \boldsymbol{(-5, 2)} \).