Question

what is pq?
learn with an example or watch a video
pq =
submit
work it out

Explanation:

Step1: Use the property of tangents from a common external point

Tangents from a common external point to a circle are equal. So, \(OP = ON = 1\), \(SN=SR = 2\), \(QP=QR\).

Step2: Let \(QP=x\). Express \(QS\) in terms of \(x\)

We know that \(QS=QR + RS\). Since \(QR = QP=x\) and \(RS = 2\), \(QS=x + 2\). Also, \(OS=ON+NS=1 + 2=3\).
In right - triangle \(OQS\) (assuming \(\angle O = 90^{\circ}\) as it is a right - angled triangle with a circle inscribed in a way that suggests a right - angle at \(O\)), by the Pythagorean theorem \(OQ^{2}+OS^{2}=QS^{2}\). And \(OQ=OP + PQ=1 + x\), \(OS = 3\), \(QS=x + 2\).
Substitute into the Pythagorean theorem: \((x + 1)^{2}+3^{2}=(x + 2)^{2}\).
Expand the equations: \(x^{2}+2x + 1+9=x^{2}+4x+4\).
Simplify the equation: \(x^{2}+2x + 10=x^{2}+4x+4\).
Subtract \(x^{2}\) from both sides: \(2x+10 = 4x+4\).
Move the \(x\) terms to one side: \(10 - 4=4x-2x\).
So, \(2x=6\), \(x = 3\).

Answer:

\(3\)