Question

given the circle below with secants ( overline{mno} ) and ( overline{qpo} ). if ( no = 7 ), ( po = 6 ) and ( qp ) is 3 more than ( mn ), find the length of ( overline{mn} ). round to the nearest tenth if necessary.

Explanation:

Step1: Apply the secant - secant theorem

The secant - secant theorem states that if two secants \(\overline{MNO}\) and \(\overline{QPO}\) are drawn to a circle from an external point \(O\), then \(MO\times NO=QO\times PO\). Let \(MN = x\). Then \(QP=x + 3\), \(MO=MN+NO=x + 7\), and \(QO=QP+PO=(x + 3)+6=x + 9\).
Substitute these values into the secant - secant formula: \((x + 7)\times7=(x + 9)\times6\).

Step2: Expand the equation

Expand both sides of the equation:

$$
LATEXBLOCK0
$$

Step3: Solve for \(x\)

Subtract \(6x\) from both sides of the equation \(7x+49 = 6x + 54\):

$$
LATEXBLOCK1
$$

Answer:

The length of \(\overline{MN}\) is \(5\).