Question

given: pq || bc. find the length of aq.

Explanation:

Step1: Apply the basic proportionality theorem (Thales' theorem)

Since \(PQ\parallel BC\), we have \(\frac{AP}{PB}=\frac{AQ}{QC}\). Let \(AQ = x\), then \(QC=40 - x\), \(AP = 6\), and \(PB = 30\). So, \(\frac{6}{30}=\frac{x}{40 - x}\).

Step2: Cross - multiply and solve the equation

Cross - multiplying gives \(6\times(40 - x)=30x\).
Expanding: \(240-6x = 30x\).
Adding \(6x\) to both sides: \(240=30x + 6x\).
Combining like terms: \(240 = 36x\).
Dividing both sides by 36: \(x=\frac{240}{36}=\frac{20}{3}\approx6.67\) (This is wrong, let's start over with correct ratio)

Wait, actually, the correct ratio is \(\frac{AP}{AB}=\frac{AQ}{AC}\) (since \(PQ\parallel BC\), triangles \(APQ\) and \(ABC\) are similar). \(AB=AP + PB=6 + 30=36\), \(AC = AQ+QC\), let \(AQ=x\), \(AC=x + 40\). So \(\frac{6}{36}=\frac{x}{x + 40}\)

Cross - multiply: \(6\times(x + 40)=36x\)

Expand: \(6x+240 = 36x\)

Subtract \(6x\) from both sides: \(240=30x\)

Divide by 30: \(x = 8\)

Answer:

\(8\)