Question

200
140
91
130
170
?
u
t
\triangle cde\sim\triangle cts

Explanation:

Step1: Identify similar triangles

Since \( QP\parallel IS\), \(\triangle UQP\sim\triangle UIS\) (by the AA similarity criterion, as corresponding angles are equal).

Step2: Set up the proportion

For similar triangles \(\triangle UQP\) and \(\triangle UIS\), the ratio of corresponding sides is equal. Let \(UQ = x\). Then \(\frac{UQ}{UI}=\frac{QP}{IS}\). We know \(UI = 170\), \(QP = 140\), \(IS = 200\), and \(UQ=x\), \(UI=x+(170 - x)\) (but more simply \(\frac{UQ}{UI}=\frac{QP}{IS}\) can be written as \(\frac{UQ}{UQ+(170 - UQ)}=\frac{140}{200}\). Cross - multiply: \(200\times UQ=140\times UI\). Also, note that the ratio of sides of similar triangles \(\frac{UQ}{UI}=\frac{140}{200}=\frac{7}{10}\). Let \(UQ = y\), \(UI=y + (170 - y)\) (another approach: since \(\frac{UQ}{UI}=\frac{QP}{IS}\), and \(UI = 170\) (assuming \(UI\) is the length from \(U\) to \(I\) along the side). Let \(UQ\) be the unknown. We can use the proportion \(\frac{UQ}{170}=\frac{140}{200}\) (because in similar triangles \(\triangle UQP\) and \(\triangle UIS\), \(\frac{\text{side of smaller triangle}}{\text{side of larger triangle}}\) for corresponding sides.

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Answer:

\(119\)