Question

find the missing length. the triangles in each pair are similar. 9. image of similar triangles 10. $\triangle wvu \sim \triangle wed$ image of similar triangles with lengths 104, 27, 117,?

Explanation:

Step1: Identify Similar Triangles Property

Since \(\triangle WVU \sim \triangle WED\), their corresponding sides are proportional. So, \(\frac{WV}{WE}=\frac{WU}{WD}=\frac{VU}{ED}\). We use the sides involving \(WV = 104\), \(WE=?\) (let's call it \(x\)), \(WU = 117\), and \(WD = 27 + x\)? Wait, no, looking at the diagram, the segments are \(WV = 104\), \(WE = x\) (the missing length), \(WU = 117\), and \(WD = 27\)? Wait, maybe the correct proportion is \(\frac{WV}{WE}=\frac{WU}{WD}\)? Wait, no, let's re - examine. The triangles are similar, so the ratio of corresponding sides should be equal. Let's assume the correspondence is \(W ightarrow W\), \(V ightarrow E\), \(U ightarrow D\). So \(\frac{WV}{WE}=\frac{WU}{WD}\). Wait, \(WV = 104\), \(WE=x\), \(WU = 117\), \(WD=27 + x\)? No, maybe the segments are \(WV = 104\), \(WE=x\), \(WU = 117\), and \(WD = 27\)? Wait, the diagram shows \(WV = 104\), \(WU = 117\), \(WD = 27\), and \(WE=x\) (the missing side). Since \(\triangle WVU\sim\triangle WED\), the ratio of \(WV\) to \(WE\) should be equal to the ratio of \(WU\) to \(WD\)? Wait, no, maybe it's \(\frac{WV}{WE}=\frac{WU}{WD}\). Wait, \(WV = 104\), \(WE=x\), \(WU = 117\), \(WD = 27\)? That doesn't make sense. Wait, maybe the correct proportion is \(\frac{WV}{WE}=\frac{WU}{WD}\) where \(WV = 104\), \(WE=x\), \(WU = 117\), and \(WD=27 + x\)? No, perhaps I misread the diagram. Let's try another approach. The formula for similar triangles is \(\frac{\text{Side of first triangle}}{\text{Corresponding side of second triangle}}=\text{constant}\). Let's take the known sides: \(WV = 104\), \(WU = 117\), \(WD = 27\), and \(WE=x\). Wait, maybe the correct proportion is \(\frac{WV}{WE}=\frac{WU}{WD}\). Wait, \(104/x=117/(27 + x)\)? No, that might be wrong. Wait, maybe the segments are \(WV = 104\), \(WE=x\), \(WU = 117\), and \(WD = 27\), and the ratio is \(\frac{WV}{WE}=\frac{WU}{WD}\), so \(104/x = 117/27\)? No, \(117/27=\frac{13}{3}\), \(104\div(13/3)=104\times\frac{3}{13}=24\). Wait, that gives \(x = 24\)? Wait, no, let's check again.

Wait, the problem is about similar triangles \(\triangle WVU\sim\triangle WED\). So the ratio of corresponding sides: \(\frac{WV}{WE}=\frac{WU}{WD}\). Let \(WE = x\). Then \(\frac{104}{x}=\frac{117}{27 + x}\)? No, cross - multiply: \(104(27 + x)=117x\), \(104\times27+104x = 117x\), \(2808=13x\), \(x = 216\)? That can't be. Wait, maybe the correct proportion is \(\frac{WV}{WE}=\frac{WU}{WD}\) where \(WV = 104\), \(WE=x\), \(WU = 117\), and \(WD = 27\). Then \(\frac{104}{x}=\frac{117}{27}\), \(117x=104\times27\), \(x=\frac{104\times27}{117}\). Simplify: \(104 = 8\times13\), \(117 = 9\times13\), \(27 = 3\times9\). So \(x=\frac{8\times13\times3\times9}{9\times13}=24\). Ah, that works. So \(\frac{WV}{WE}=\frac{WU}{WD}\) with \(WV = 104\), \(WE=x\), \(WU = 117\), \(WD = 27\) (wait, no, \(WD\) should be corresponding to \(WU\)? Maybe the correct correspondence is \(WV\) corresponds to \(WE\), \(WU\) corresponds to \(WD\), but \(WD\) is \(27\)? No, maybe the diagram has \(WD = 27\) and \(WU = 117\), \(WV = 104\), \(WE=x\). So using the proportion \(\frac{WV}{WE}=\frac{WU}{WD}\), we have \(\frac{104}{x}=\frac{117}{27}\). Wait, \(117\div27=\frac{13}{3}\), \(104\div\frac{13}{3}=104\times\frac{3}{13}=24\).

Step2: Solve for the Missing Length

We set up the proportion based on the similarity of triangles:
\(\frac{WV}{WE}=\frac{WU}{WD}\)
Substitute the known values \(WV = 104\), \(WU = 117\), \(WD = 27\) and let \(WE=x\):
\(\frac{104}{x}=\frac{117}{27}\)
Cross - multiply: \(117x=104\times27\)
Simplify the right - hand side: \(104\times27 = 2808\)
Then \(x=\frac{2808}{117}\)
Simplify the fraction: Divide numerator and denominator by 13: \(2808\div13 = 216\), \(117\div13 = 9\). So \(\frac{216}{9}=24\)

Answer:

24