Question

find hi.
hi =
submit
work it out

Explanation:

Step1: Identify Similar Triangles

The segments with arrows are midsegments (or the triangles are similar by the Basic Proportionality Theorem/Thales' theorem). So, $\triangle JHI$ and $\triangle JHF$ are similar? Wait, actually, the line $IG$ is parallel to $JF$? Wait, no, the markings: the two red arrows on $JI$ and $IG$? Wait, no, looking at the diagram, $G$ is on $FH$, and $I$ is on $JH$, with $IG$ parallel to $JF$ (since the arrows indicate midsegments or parallel lines with proportional segments). So, the ratio of $HG$ to $HF$ is $13$ to $39$, which simplifies to $\frac{13}{39}=\frac{1}{3}$? Wait, no, $HG = 13$, $HF = HG + GF = 13 + GF$? Wait, no, the vertical segment is $39$ (from $H$ to the top), and $HG$ is $13$, so $GF = 39 - 13 = 26$? Wait, no, the diagram shows that the length from $H$ to the top (let's say $F$'s vertical line) is $39$, and $HG$ is $13$, so $HG:HF = 13:39 = 1:3$? Wait, no, $HF$ is the entire length, so $HG$ is part of $HF$. Wait, actually, the triangles: $\triangle JIG$ and $\triangle JHF$? No, better: since $IG$ is parallel to $JF$ (by the midsegment theorem, as the arrows on $JI$ and $JG$? Wait, the red arrows are on $JI$ (wait, $JI$ is $36$, and $IH$ is what we need to find. Wait, the key is that the line $IG$ is parallel to $JF$, so by the Basic Proportionality Theorem (Thales' theorem), $\frac{JI}{JH}=\frac{HG}{HF}$. Wait, $JH = JI + IH$, let $IH = x$, so $JH = 36 + x$. $HG = 13$, $HF = 39$ (since the vertical segment is $39$). Wait, no, $HF$ is the length from $H$ to $F$, which is $39$? Wait, the diagram shows the vertical line (from $H$ up) is $39$, and $HG$ is $13$, so $HG/HF = 13/39 = 1/3$. Then, by Thales' theorem, since $IG \parallel JF$, $\frac{JI}{JH} = \frac{HG}{HF}$. Wait, $JI = 36$, $JH = JI + IH = 36 + x$, $HG = 13$, $HF = 39$. So:

$\frac{JI}{JH} = \frac{HG}{HF}$

$\frac{36}{36 + x} = \frac{13}{39}$

Simplify $\frac{13}{39} = \frac{1}{3}$

So:

$\frac{36}{36 + x} = \frac{1}{3}$

Cross-multiplying:

$36 \times 3 = 36 + x$

$108 = 36 + x$

Subtract $36$:

$x = 108 - 36 = 72$? Wait, that can't be, because $JI$ is $36$, and $IH$ would be $72$, making $JH = 108$. But let's check the ratio again. Wait, maybe I got the ratio reversed. Maybe $\frac{IH}{JI} = \frac{HG}{GF}$? Wait, $GF = 39 - 13 = 26$, so $HG:GF = 13:26 = 1:2$. Then, by the midsegment theorem, if $IG$ is parallel to $JF$, then $IH:JI = HG:GF = 1:2$. So $IH = \frac{1}{2} JI = \frac{1}{2} \times 36 = 18$? Wait, that makes more sense. Wait, let's re-examine:

If $IG$ is parallel to $JF$, then the triangles $\triangle HIG$ and $\triangle HJF$ are similar. So the ratio of corresponding sides is equal. So $\frac{IH}{JH} = \frac{HG}{HF}$. Wait, $JH = JI + IH = 36 + IH$, $HF = 39$, $HG = 13$. So $\frac{IH}{36 + IH} = \frac{13}{39} = \frac{1}{3}$. Then:

$3 \times IH = 36 + IH$

$3IH - IH = 36$

$2IH = 36$

$IH = 18$

Ah, that's better. So the ratio is $\frac{HG}{HF} = \frac{13}{39} = \frac{1}{3}$, so the smaller triangle ( $\triangle HIG$ ) is similar to the larger triangle ( $\triangle HJF$ ) with ratio $1:3$. Therefore, $\frac{IH}{JH} = \frac{1}{3}$, and $JH = JI + IH = 36 + IH$. So:

$\frac{IH}{36 + IH} = \frac{1}{3}$

Multiply both sides by $3(36 + IH)$:

$3IH = 36 + IH$

Subtract $IH$:

$2IH = 36$

Divide by $2$:

$IH = 18$

Step2: Verify the Ratio

Check the ratio of $HG$ to $HF$: $13/39 = 1/3$. The ratio of $IH$ to $JH$: $18/(36 + 18) = 18/54 = 1/3$. So that matches. So the length of $HI$ is $18$.

Answer:

$\boxed{18}$