Question

1. $\triangle rst \sim \triangle ysz$; find $yz$

(image of triangle rst with y on rs, z on st, yz labeled $2x + 2$, rt labeled $3x - 7$, sz segment with length related to 40 and zt segment with length 5)

Explanation:

Step1: State the similarity ratio

Since \(\triangle RST \sim \triangle YSZ\), the ratios of corresponding sides are equal. The ratio of \(SZ\) to \(ST\) is \(\frac{SZ}{ST}=\frac{40}{40 + 5}=\frac{40}{45}=\frac{8}{9}\)? Wait, no, wait. Wait, actually, \(SZ\) and \(ZT\): Wait, looking at the diagram, \(ST = SZ+ZT\)? Wait, no, the segments: \(SZ\) is part of \(ST\)? Wait, no, the labels: \(Y\) is on \(RS\), \(Z\) is on \(ST\)? Wait, no, the triangle is \(RST\), with \(Y\) on \(RS\) and \(Z\) on \(ST\), and \(YZ\) parallel to \(RT\)? Wait, no, the similarity is \(\triangle RST \sim \triangle YSZ\), so corresponding sides: \(YZ\) corresponds to \(RT\), and \(SZ\) corresponds to \(ST\). Wait, \(ST\) is \(SZ + ZT\)? Wait, the lengths: \(SZ\) is 40? Wait, no, the diagram has \(SZ\) as 40? Wait, no, the arrow marks: the segment from \(S\) to the top arrow is 40, and from \(Z\) to \(T\) is 5, so \(ST = SZ + ZT = 40 + 5 = 45\)? Wait, no, maybe \(SZ\) is 40, and \(ZT\) is 5, so \(ST = SZ + ZT = 45\). Then \(YZ\) is \(2x + 2\), \(RT\) is \(3x - 7\), and \(SZ\) is 40, \(ST\) is 45? Wait, no, maybe the ratio is \(\frac{YZ}{RT}=\frac{SZ}{ST}\). Wait, \(\triangle YSZ \sim \triangle RST\), so corresponding sides: \(YS\) corresponds to \(RS\), \(SZ\) corresponds to \(ST\), \(YZ\) corresponds to \(RT\). So the ratio of similarity is \(\frac{SZ}{ST}=\frac{40}{40 + 5}=\frac{40}{45}=\frac{8}{9}\)? Wait, no, that can't be. Wait, maybe \(ZT = 5\) and \(SZ = 40\), so \(ST = SZ + ZT = 45\), and \(YZ\) is parallel to \(RT\), so by the Basic Proportionality Theorem (Thales' theorem), but since the triangles are similar, the ratio of \(YZ\) to \(RT\) is equal to the ratio of \(SZ\) to \(ST\). Wait, no, \(\triangle YSZ \sim \triangle RST\), so \(\frac{YZ}{RT}=\frac{SZ}{ST}\). So \(YZ = 2x + 2\), \(RT = 3x - 7\), \(SZ = 40\), \(ST = SZ + ZT = 40 + 5 = 45\). So \(\frac{2x + 2}{3x - 7}=\frac{40}{45}\). Simplify \(\frac{40}{45}=\frac{8}{9}\). So \(\frac{2x + 2}{3x - 7}=\frac{8}{9}\). Cross - multiply: \(9(2x + 2)=8(3x - 7)\).

Step2: Solve the equation

Expand both sides: \(18x+18 = 24x - 56\).
Subtract \(18x\) from both sides: \(18=6x - 56\).
Add 56 to both sides: \(18 + 56=6x\), so \(74 = 6x\)? Wait, that can't be. Wait, maybe I got the ratio wrong. Maybe \(SZ\) is 40, and \(ZT\) is 5, so \(ST = 5\) and \(SZ = 40\)? No, that doesn't make sense. Wait, maybe the ratio is \(\frac{SZ}{ZT}=\frac{YZ}{RT}\)? No, similarity of triangles: corresponding sides. Wait, maybe \(\triangle RST \sim \triangle YSZ\), so the order is \(R\) corresponds to \(Y\), \(S\) corresponds to \(S\), \(T\) corresponds to \(Z\). Wait, that would mean \(RS\) corresponds to \(YS\), \(ST\) corresponds to \(SZ\), \(RT\) corresponds to \(YZ\). So then \(\frac{RT}{YZ}=\frac{ST}{SZ}\). So \(RT = 3x - 7\), \(YZ = 2x + 2\), \(ST = SZ + ZT = 40 + 5 = 45\), \(SZ = 40\). So \(\frac{3x - 7}{2x + 2}=\frac{45}{40}=\frac{9}{8}\). Cross - multiply: \(8(3x - 7)=9(2x + 2)\).

Step3: Solve the correct equation

Expand: \(24x-56 = 18x + 18\).
Subtract \(18x\) from both sides: \(6x-56 = 18\).
Add 56 to both sides: \(6x=18 + 56=74\)? No, that's not right. Wait, maybe the segments are \(SZ = 40\) and \(ZT = 5\), so \(ST = SZ + ZT = 45\), and \(YZ\) is parallel to \(RT\), so by the Basic Proportionality Theorem (Thales' theorem), \(\frac{RY}{YS}=\frac{ZT}{SZ}\), but no, the triangles are similar. Wait, maybe the ratio is \(\frac{YZ}{RT}=\frac{SZ}{ST}\), where \(ST = SZ + ZT = 40 + 5 = 45\), \(SZ = 40\), so \(\frac{2x + 2}{3x - 7}=\frac{40}{45}=\frac{8}{9}\). Then cross - multiply: \(9(2x + 2)=8(3x - 7)\).

Step4: Correct equation solving

\(18x + 18 = 24x - 56\)
Subtract \(18x\) from both sides: \(18 = 6x - 56\)
Add 56 to both sides: \(6x = 18 + 56 = 74\)? No, that gives \(x=\frac{74}{6}=\frac{37}{3}\), which is not an integer. That can't be. Wait, maybe I misread the diagram. Maybe \(SZ = 40\) is not, but \(ST = 40\) and \(ZT = 5\), so \(SZ = ST - ZT = 40 - 5 = 35\). Ah! That's probably the mistake. So \(ST = 40\), \(ZT = 5\), so \(SZ = ST - ZT = 40 - 5 = 35\). Then the ratio of \(SZ\) to \(ST\) is \(\frac{SZ}{ST}=\frac{35}{40}=\frac{7}{8}\)? No, wait, \(\triangle RST \sim \triangle YSZ\), so \(YZ\) corresponds to \(RT\), \(SZ\) corresponds to \(ST\). So \(SZ = 35\), \(ST = 40\), \(YZ = 2x + 2\), \(RT = 3x - 7\). So \(\frac{YZ}{RT}=\frac{SZ}{ST}\), so \(\frac{2x + 2}{3x - 7}=\frac{35}{40}=\frac{7}{8}\). Cross - multiply: \(8(2x + 2)=7(3x - 7)\).

Step5: Solve the new equation

\(16x + 16 = 21x - 49\)
Subtract \(16x\) from both sides: \(16 = 5x - 49\)
Add 49 to both sides: \(5x = 16 + 49 = 65\)
Divide by 5: \(x = 13\)

Step6: Find \(YZ\)

Now that \(x = 13\), substitute into \(YZ = 2x + 2\): \(YZ = 2(13)+2 = 26 + 2 = 28\). Wait, let's check with \(RT = 3x - 7 = 3(13)-7 = 39 - 7 = 32\). Then the ratio of \(YZ\) to \(RT\) is \(\frac{28}{32}=\frac{7}{8}\), and the ratio of \(SZ\) to \(ST\) is \(\frac{35}{40}=\frac{7}{8}\). Yes, that works. So the mistake was in the length of \(ST\): I thought \(SZ\) was 40, but actually \(ST\) is 40, and \(ZT\) is 5, so \(SZ = ST - ZT = 40 - 5 = 35\).

Answer:

\(YZ = 28\)