Question

graded quiz: circles
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in the above figure, whats the length of the tangent from the external point d to point b?

Explanation:

Step1: Apply the tangent - secant theorem

The tangent - secant theorem states that if a tangent segment \(TD\) and a secant segment \(TAC\) are drawn to a circle from an external point \(T\), then \(TD^{2}=TA\times TC\). Here, the external point is \(D\), \(TA = AB + BC+CD=3 + 2+2=7\), \(TC=BC + CD=2 + 2 = 4\)

Step2: Calculate the length of the tangent

Let the length of the tangent from \(D\) to \(B\) be \(x\). Using the formula \(x^{2}=CD\times (AB + BC+CD)\) (by the tangent - secant relationship, where if we consider the secant \(AD\) and tangent \(DB\)). Substitute \(AB = 3\), \(BC = 2\), \(CD=2\) into the formula.

We have \(x^{2}=2\times(3 + 2+2)\)

\(x^{2}=2\times7\)

\(x^{2}=14\)

\(x=\sqrt{14}\)

Answer:

\(\sqrt{14}\)