Question

find hi.
hi =
submit

Explanation:

Step1: Identify Similar Triangles

The lines \( GJ \) and \( HI \) are parallel (indicated by the arrows), so triangles \( KHJ \) and \( KHI \) are similar by the Basic Proportionality Theorem (Thales' theorem). Thus, the ratios of corresponding sides are equal: \(\frac{KH}{KG}=\frac{JH}{IH}\). Wait, actually, \( KH = 66 \), \( GH = 36 \), so \( KG=KH - GH=66 - 36 = 30 \)? Wait, no, let's re - check. The length of \( KH \) is 66, and \( GH = 36 \), so \( KG=KH - GH = 66-36 = 30 \)? Wait, no, maybe the sides: \( KH = 66 \), \( GH = 36 \), so \( KG=KH - GH=66 - 36 = 30 \)? Wait, the vertical side: \( JH = 25 + HI \), and the triangles \( KJH \) and \( KIH \) are similar? Wait, no, the segments \( GJ \) and \( HI \) are parallel, so triangle \( KJG \) and triangle \( KIH \) are similar? Wait, let's use the basic proportionality. The line \( GI \) (wait, the midline? No, the segments with arrows: the two lines are parallel, so by the converse of the basic proportionality theorem, the triangles are similar. Let's denote \( HI = x \), \( JI = 25 \), so \( JH=x + 25 \). The base \( KH = 66 \), \( GH = 36 \), so \( KG=KH - GH=66 - 36 = 30 \). So the ratio of the bases \( KG/KH=30/66 = 5/11 \)? Wait, no, maybe the other way: the ratio of the segments on the base: \( GH/KH = 36/66=6/11 \), and the ratio of the segments on the height: \( HI/JH=x/(x + 25) \). Since the lines are parallel, the triangles are similar, so the ratios of corresponding sides are equal. So \( \frac{GH}{KH}=\frac{HI}{JH} \). Wait, \( GH = 36 \), \( KH = 66 \), \( JH=HI + 25 \). So \( \frac{36}{66}=\frac{HI}{HI + 25} \). Cross - multiply: \( 36(HI + 25)=66HI \). Expand: \( 36HI+900 = 66HI \). Subtract \( 36HI \): \( 900 = 30HI \). Then \( HI=\frac{900}{30}=30 \)? Wait, no, that can't be. Wait, maybe I mixed up the triangles. Let's look again. The length of \( KH \) is 66, and \( GH = 36 \), so \( KG = KH - GH=66 - 36 = 30 \). So the ratio of \( KG \) to \( KH \) is \( 30/66 = 5/11 \)? No, \( 30/66=5/11 \), \( 36/66 = 6/11 \). Wait, the vertical side: \( JH \) is the total height, and \( HI \) is the height of the smaller triangle. So by similar triangles, \( \frac{KG}{KH}=\frac{HI}{JH} \). \( KG = 30 \), \( KH = 66 \), \( JH=HI + 25 \). So \( \frac{30}{66}=\frac{HI}{HI + 25} \). Cross - multiply: \( 30(HI + 25)=66HI \). \( 30HI+750 = 66HI \). \( 750 = 36HI \). \( HI=\frac{750}{36}=\frac{125}{6}\approx20.83 \)? No, that's wrong. Wait, maybe the triangles are \( KJH \) and \( GIH \). Wait, \( KH = 66 \), \( GH = 36 \), so \( KG = 66 - 36 = 30 \). \( JH = 25 + HI \), \( IH = HI \). Since \( GJ\parallel HI \), triangle \( KJH\sim triangle KIH \)? No, \( GJ\parallel HI \), so angle \( K \) is common, and angle \( KGI = angle KJH \) (corresponding angles), so triangle \( KGI\sim triangle KJH \). So \( \frac{KG}{KH}=\frac{GI}{JH} \). Wait, \( GI = HI \), \( JH=HI + 25 \), \( KG = 30 \), \( KH = 66 \). So \( \frac{30}{66}=\frac{HI}{HI + 25} \). Cross - multiply: \( 30(HI + 25)=66HI \). \( 30HI+750 = 66HI \). \( 750 = 36HI \). \( HI=\frac{750}{36}=\frac{125}{6}\approx20.83 \). No, that's not right. Wait, maybe the other way: the ratio of the bases is \( GH/KH = 36/66 = 6/11 \), so the ratio of the heights should be the same. So \( HI/JH=6/11 \), and \( JH = HI + 25 \). So \( HI/(HI + 25)=6/11 \). Cross - multiply: \( 11HI=6(HI + 25) \). \( 11HI=6HI + 150 \). \( 5HI = 150 \). \( HI = 30 \). Wait, that's different. Wait, \( 11HI-6HI = 150 \), \( 5HI = 150 \), \( HI = 30 \). But then \( JH=30 + 25 = 55 \), and \( 30/55 = 6/11 \), and \( 36/66 = 6/11 \). Yes! That works. So I made a mistake earlier in the \( KG \) calculation. The correct ratio is \( GH/KH = 36/66 = 6/11 \), so \( HI/JH = 6/11 \), and \( JH=HI + 25 \). So solving \( HI/(HI + 25)=6/11 \):

Step1: Set up the proportion

Since the two lines are parallel, the triangles are similar, so the ratio of the corresponding sides is equal. Let \( HI=x \), then \( JH=x + 25 \). The ratio of the bases \( \frac{GH}{KH}=\frac{36}{66}=\frac{6}{11} \), and the ratio of the heights \( \frac{HI}{JH}=\frac{x}{x + 25} \). So we have the equation:
\[ \frac{x}{x + 25}=\frac{6}{11} \]

Step2: Solve the proportion

Cross - multiply:
\[ 11x=6(x + 25) \]
Expand the right - hand side:
\[ 11x=6x+150 \]
Subtract \( 6x \) from both sides:
\[ 11x - 6x=6x + 150-6x \]
\[ 5x = 150 \]
Divide both sides by 5:
\[ x=\frac{150}{5}=30 \]

Answer:

\( HI = 30 \)