the medians of △jkl are overline{jn}, overline{kp}, and overline{lm}. they meet at a single point q. (in other words, q is the centroid of △jkl.) suppose kp = 36, jq = 22, and lq = 16. find the following lengths. note that the figure is not drawn to scale.

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Accepted Answer

# Explanation:
## Step1: Recall Centroid Property
The centroid of a triangle divides each median into a ratio of $2:1$, with the longer segment being closer to the vertex. So, for any median, the length from the vertex to the centroid is twice the length from the centroid to the midpoint of the side.

## Step2: Find $QP$
Given $KP = 36$. Let $QP=x$ and $KQ = 2x$ (since centroid divides median in $2:1$). Then $KQ+QP=KP$, so $2x + x=36$, $3x = 36$, $x=\frac{36}{3}=12$. Thus, $QP = 12$.

## Step3: Find $QM$
Given $LQ = 16$. Let $QM = y$ and $LQ=2y$? Wait, no: $LQ$ is from $L$ to $Q$, and $LM$ is the median. So $LQ$ is the segment from vertex $L$ to centroid $Q$, so $LQ = 2 \times QM$. So $16 = 2 \times QM$, then $QM=\frac{16}{2}=8$? Wait, no, wait: Wait, the median is $LM$, so $L$ to $M$ (midpoint of $JK$). So $Q$ is centroid, so $LQ:QM = 2:1$. So $LQ = 2 \times QM$, so $QM=\frac{LQ}{2}=\frac{16}{2}=8$? Wait, no, wait the problem says $LQ = 16$. Wait, maybe I mixed up. Wait, the median is $LM$, so $M$ is midpoint of $JK$, so $LM$ is median, with $Q$ on $LM$. So $LQ$ is from $L$ to $Q$, $QM$ is from $Q$ to $M$. So centroid ratio: $LQ:QM = 2:1$, so $LQ = 2 \times QM$, so $QM=\frac{LQ}{2}=\frac{16}{2}=8$. Wait, but let's check the other median $JN$.

## Step4: Find $JN$
Given $JQ = 22$. $JN$ is the median from $J$ to $N$ (midpoint of $KL$). So $JQ$ is from $J$ to $Q$, $QN$ is from $Q$ to $N$. Centroid ratio: $JQ:QN = 2:1$, so $JN = JQ + QN = JQ+\frac{JQ}{2}=22 + 11 = 33$. Wait, because $JQ = 2 \times QN$, so $QN=\frac{22}{2}=11$, so $JN=22 + 11 = 33$.

Wait, let's re-express:

For median $KP$ (from $K$ to $P$, midpoint of $JL$): $KQ:QP = 2:1$, so $KP = KQ + QP = 2QP + QP = 3QP$, so $QP=\frac{KP}{3}=\frac{36}{3}=12$ (which matches step 2).

For median $JN$ (from $J$ to $N$, midpoint of $KL$): $JQ:QN = 2:1$, so $JN = JQ + QN = JQ+\frac{JQ}{2}=\frac{3JQ}{2}$. Since $JQ = 22$, $JN=\frac{3 \times 22}{2}=33$.

For median $LM$ (from $L$ to $M$, midpoint of $JK$): $LQ:QM = 2:1$, so $LM = LQ + QM = LQ+\frac{LQ}{2}=\frac{3LQ}{2}$, but we need $QM$. Since $LQ = 16$, $QM=\frac{LQ}{2}=8$ (because $LQ = 2 \times QM$).

Wait, let's confirm each:

1. $QP$: Median $KP$, centroid divides into $2:1$, so total parts $3$. $KP = 36$, so each part $12$. So $QP = 12$ (since $KQ = 24$, $QP = 12$, $24 + 12 = 36$).

2. $QM$: Median $LM$, $LQ = 16$ (from $L$ to $Q$), so $QM$ is half of $LQ$ (since ratio $2:1$), so $QM = 8$.

3. $JN$: Median $JN$, $JQ = 22$ (from $J$ to $Q$), so $QN$ is half of $JQ$ (ratio $2:1$), so $QN = 11$, so $JN = JQ + QN = 22 + 11 = 33$.

Wait, but let's check the centroid property again: The centroid divides each median into a ratio of $2:1$, where the length from the vertex to the centroid is twice the length from the centroid to the midpoint of the side. So for median $KP$ (vertex $K$, midpoint $P$ of $JL$): $KQ:QP = 2:1$, so $KQ = 2 \times QP$, and $KP = KQ + QP = 3 \times QP$, so $QP = \frac{KP}{3} = \frac{36}{3} = 12$. Correct.

For median $JN$ (vertex $J$, midpoint $N$ of $KL$): $JQ:QN = 2:1$, so $JQ = 2 \times QN$, and $JN = JQ + QN = 3 \times QN$, or $JN = \frac{3}{2} \times JQ$ (since $JQ = 2 \times QN$, so $QN = \frac{JQ}{2}$, so $JN = JQ + \frac{JQ}{2} = \frac{3JQ}{2} = \frac{3 \times 22}{2} = 33$. Correct.

For median $LM$ (vertex $L$, midpoint $M$ of $JK$): $LQ:QM = 2:1$, so $LQ = 2 \times QM$, so $QM = \frac{LQ}{2} = \frac{16}{2} = 8$. Correct.

So:

- $QP = 12$

- $QM = 8$

- $JN = 33$

# Answer:
$JN = 33$, $QM = 8$, $QP = 12$

(Assuming the question is to find $JN$, $QM$, $QP$ as per the boxes. So:

$JN = 33$, $QM = 8$, $QP = 12$)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall Centroid Property

Step2: Find \(QP\)

Step3: Find \(QM\)

Step4: Find \(JN\)

Answer: