Question

find the exact length of the midsegment of trapezoid jklm with vertices j(6, 10), k(10, 6), l(8, 2), and m(2, 2). the length of the midsegment is \square.

Explanation:

Step1: Identify the bases of the trapezoid

First, we need to determine which sides are parallel (the bases). Let's find the slopes of the sides.
The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Slope of $JM$: $J(6,10)$, $M(2,2)$
$m_{JM} = \frac{2 - 10}{2 - 6} = \frac{-8}{-4} = 2$

Slope of $KL$: $K(10,6)$, $L(8,2)$
$m_{KL} = \frac{2 - 6}{8 - 10} = \frac{-4}{-2} = 2$

Slope of $JK$: $J(6,10)$, $K(10,6)$
$m_{JK} = \frac{6 - 10}{10 - 6} = \frac{-4}{4} = -1$

Slope of $ML$: $M(2,2)$, $L(8,2)$
$m_{ML} = \frac{2 - 2}{8 - 2} = \frac{0}{6} = 0$

Since the slopes of $JM$ and $KL$ are equal (both 2), $JM$ and $KL$ are parallel (the bases). Now we find the lengths of $JM$ and $KL$.

Step2: Calculate the length of $JM$

Using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for $J(6,10)$ and $M(2,2)$:
$d_{JM} = \sqrt{(2 - 6)^2 + (2 - 10)^2} = \sqrt{(-4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$

Step3: Calculate the length of $KL$

Using the distance formula for $K(10,6)$ and $L(8,2)$:
$d_{KL} = \sqrt{(8 - 10)^2 + (2 - 6)^2} = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

Step4: Find the length of the midsegment

The length of the midsegment of a trapezoid is the average of the lengths of the two bases. The formula is $m = \frac{d_1 + d_2}{2}$, where $d_1$ and $d_2$ are the lengths of the bases.
So, $m = \frac{4\sqrt{5} + 2\sqrt{5}}{2} = \frac{6\sqrt{5}}{2} = 3\sqrt{5}$

Wait, that seems incorrect. Wait, maybe I made a mistake in identifying the bases. Let's check the other sides. Wait, the slope of $ML$ is 0, so it's horizontal. Let's check the y-coordinates of $M(2,2)$ and $L(8,2)$: they have the same y-coordinate, so $ML$ is horizontal. Now check the slope of $JK$: -1, and slope of $JM$: 2, slope of $KL$: 2. Wait, maybe $ML$ and $JK$ are not the bases. Wait, let's re-examine the coordinates.

Points: $J(6,10)$, $K(10,6)$, $L(8,2)$, $M(2,2)$

Let's plot them mentally: $M(2,2)$ and $L(8,2)$ are on the horizontal line $y = 2$. $J(6,10)$ and $K(10,6)$: let's see the y-coordinates. Now, the midsegment of a trapezoid is parallel to the bases and its length is the average of the lengths of the two bases.

Wait, maybe the bases are $ML$ and $JK$? No, $ML$ is horizontal (slope 0), $JK$ has slope -1. Not parallel. Wait, $ML$ is from (2,2) to (8,2), length is $8 - 2 = 6$ (since y is constant). $JM$: from (6,10) to (2,2), length we calculated as $4\sqrt{5} \approx 8.94$. $KL$: from (10,6) to (8,2), length $2\sqrt{5} \approx 4.47$. $JK$: from (6,10) to (10,6), length $\sqrt{(10 - 6)^2 + (6 - 10)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \approx 5.66$.

Wait, maybe I made a mistake in slope calculation. Let's recalculate slopes:

Slope of $ML$: $M(2,2)$, $L(8,2)$: $\frac{2 - 2}{8 - 2} = 0$, correct.

Slope of $JK$: $J(6,10)$, $K(10,6)$: $\frac{6 - 10}{10 - 6} = \frac{-4}{4} = -1$, correct.

Slope of $JM$: $J(6,10)$, $M(2,2)$: $\frac{2 - 10}{2 - 6} = \frac{-8}{-4} = 2$, correct.

Slope of $KL$: $K(10,6)$, $L(8,2)$: $\frac{2 - 6}{8 - 10} = \frac{-4}{-2} = 2$, correct.

So $JM$ and $KL$ are parallel (both slope 2), so they are the legs? No, legs are non-parallel. Wait, no: in a trapezoid, only one pair of sides is parallel (in some definitions, at least one pair). Wait, maybe this is a trapezoid with bases $ML$ and $JK$? No, they are not parallel. Wait, no, $ML$ is horizontal, let's check the other side: $J(6,10)$ and $K(10,6)$: not horizontal. Wait, $M(2,2)$ and $L(8,2)$: length 6 (since x from 2 to 8, y=2). $J(6,10)$ and $K(10,6)$: length $\sqrt{(10 - 6)^2 + (6 - 10)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$. Wait, no, maybe the bases are $ML$ (length 6) and $JK$ (length $4\sqrt{2}$)? But they are not parallel. Wait, I must have messed up.

Wait, let's list all sides:

1. $JM$: (6,10)-(2,2)
2. $MK$: (2,2)-(10,6) – no, the vertices are J, K, L, M, so the sides are JK, KL, LM, MJ.

Wait, the trapezoid is J-K-L-M-J. So sides:

JK: J(6,10) to K(10,6)

KL: K(10,6) to L(8,2)

LM: L(8,2) to M(2,2)

MJ: M(2,2) to J(6,10)

Now, check which sides are parallel:

Slope of JK: (6-10)/(10-6) = -4/4 = -1

Slope of KL: (2-6)/(8-10) = -4/-2 = 2

Slope of LM: (2-2)/(2-8) = 0/-6 = 0 (wait, L(8,2) to M(2,2): x from 8 to 2, so slope is (2-2)/(2-8) = 0/-6 = 0, correct.

Slope of MJ: (10-2)/(6-2) = 8/4 = 2 (wait, I had the slope formula wrong earlier! Slope is (y2 - y1)/(x2 - x1), so for MJ: M(2,2) to J(6,10), it's (10 - 2)/(6 - 2) = 8/4 = 2. Ah! I had the points reversed earlier. So slope of MJ is 2, slope of KL is 2. So KL and MJ are parallel (both slope 2). So the bases are KL and MJ. Then the other two sides are JK and LM, which are the legs (non-parallel).

Now, length of LM: L(8,2) to M(2,2): since y is constant, length is |8 - 2| = 6.

Length of JK: J(6,10) to K(10,6): $\sqrt{(10 - 6)^2 + (6 - 10)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$. Wait, no, the midsegment is parallel to the bases and its length is the average of the lengths of the two bases (KL and MJ).

Wait, let's calculate length of KL: K(10,6) to L(8,2): $\sqrt{(8 - 10)^2 + (2 - 6)^2} = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$.

Length of MJ: M(2,2) to J(6,10): $\sqrt{(6 - 2)^2 + (10 - 2)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$.

Now, the midsegment length is the average of the lengths of the two bases: (length of KL + length of MJ)/2 = (2\sqrt{5} + 4\sqrt{5})/2 = (6\sqrt{5})/2 = 3\sqrt{5} ≈ 6.708. But wait, LM is length 6 (from (8,2) to (2,2)), which is horizontal. Wait, maybe I made a mistake in identifying the bases. Let's check the y-coordinates: M(2,2) and L(8,2) are on y=2, J(6,10) and K(10,6) are above. Wait, maybe the bases are LM (length 6) and JK (length 4\sqrt{2})? No, they are not parallel. Wait, no, KL and MJ are parallel (slope 2), so they are the bases. Then the midsegment should be parallel to them and average their lengths. But let's check the other sides: JK has slope -1, LM has slope 0, so they are not parallel. So the trapezoid has bases KL (length 2\sqrt{5}) and MJ (length 4\sqrt{5}), and legs JK (length 4\sqrt{2}) and LM (length 6).

But wait, the midsegment length is also equal to the average of the lengths of the two bases, so (2\sqrt{5} + 4\sqrt{5})/2 = 3\sqrt{5}. But let's check with the other pair: if the bases were LM (length 6) and... but LM is horizontal, is there another horizontal side? No. So the correct bases are KL and MJ (parallel, slope 2). So midsegment length is 3\sqrt{5}. But wait, let's check the coordinates again.

Wait, maybe the problem is that I misidentified the sides. Let's list the vertices in order: J(6,10), K(10,6), L(8,2), M(2,2), back to J(6,10). So the sides are JK, KL, LM, MJ.

JK: J to K

KL: K to L

LM: L to M

MJ: M to J

So KL is from K(10,6) to L(8,2), MJ is from M(2,2) to J(6,10). These two have the same slope (2), so they are parallel. So the bases are KL and MJ. Then the midsegment length is (length of KL + length of MJ)/2.

Length of KL: distance between K(10,6) and L(8,2): $\sqrt{(8 - 10)^2 + (2 - 6)^2} = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

Length of MJ: distance between M(2,2) and J(6,10): $\sqrt{(6 - 2)^2 + (10 - 2)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}$

Midsegment length: (2\sqrt{5} + 4\sqrt{5})/2 = 6\sqrt{5}/2 = 3\sqrt{5}

But wait, let's check the other sides: LM is from L(8,2) to M(2,2), length is 8 - 2 = 6 (since y is constant). JK is from J(6,10) to K(10,6), length is $\sqrt{(10 - 6)^2 + (6 - 10)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

Now, the midsegment should also be equal to the average of the lengths of the two bases, which are KL and MJ. So that's 3\sqrt{5}. But let's verify with the formula for the midsegment: it's parallel to the bases and its length is the average of the two bases.

Alternatively, maybe the bases are LM (length 6) and... wait, LM is horizontal, is there another horizontal side? No, J and K are not on the same horizontal line. So the correct bases are KL and MJ, so midsegment length is 3\sqrt{5}.

But wait, let's calculate the midsegment using the coordinates of the midpoints of the legs. The legs are JK and LM.

Midpoint of JK: J(6,10) and K(10,6): midpoint is ((6 + 10)/2, (10 + 6)/2) = (8, 8)

Midpoint of LM: L(8,2) and M(2,2): midpoint is ((8 + 2)/2, (2 + 2)/2) = (5, 2)

Now, the midsegment is the segment connecting these two midpoints. Let's calculate its length.

Distance between (8,8) and (5,2): $\sqrt{(5 - 8)^2 + (2 - 8)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$. Yes! That matches. So the length of the midsegment is $3\sqrt{5}$.

Answer:

$3\sqrt{5}$