Question

find the value of n.

Explanation:

Step1: Determine similar triangles

Since \(SR\parallel SQ\) (by the given figure's parallel - like structure), \(\triangle SPR\sim\triangle QPR\) (by the Angle - Angle similarity criterion as \(\angle SPR=\angle QPR\) (given angle) and \(\angle SRP + \angle QRP = 180^{\circ}\) and the lines are parallel, so corresponding angles are equal).
The ratio of corresponding sides of similar triangles is equal. So, \(\frac{SP}{QP}=\frac{SR}{QR}\).
We know \(SP = 35\), \(SR=48 - 27=21\), \(QR = 27\). Let \(QP=n\).

Step2: Set up the proportion

Substitute the values into the proportion \(\frac{35}{n}=\frac{21}{27}\).
Cross - multiply: \(21n=35\times27\).
First, calculate \(35\times27=(30 + 5)\times27=30\times27+5\times27=810 + 135 = 945\).
So, \(21n = 945\).
Then, solve for \(n\): \(n=\frac{945}{21}\).
Since \(945\div21 = 45\) (because \(21\times45=(20 + 1)\times45=20\times45+1\times45=900+45 = 945\)).

Answer:

\(n = 45\)