Segments Formed by Two Secants

How to find the segments formed by two intersecting secants of a circle: formula, 1 example, and its solution.

Formula

Formula

PB and PD are the secants
that start from the same point P.

Then four segments are formed:
PA, AB, PC, and CD.

Then
PA⋅PB = PC⋅PD.
PA⋅(PA + AB) = PC⋅(PC + CD)

Example

Example

Solution

x, 3, 4, and 6
are the segments
formed by two intersecting secants.

So x(x + 3) = 4(4 + 6).

x(x + 3) = x2 + 3x

Multiply a Monomial and a Polynomial

4 + 6 = 10

4⋅10 = 40

Move 40 to the left side.

Factor the right side
x2 + 3x - 40.

Find a pair of numbers
whose product is the constant term -40
and whose sum is the coefficient of the middle term +3.

-5⋅8 = -40
-5 + 8 = +3

Then (x - 5)(x + 8) = 0.

Factor a Quadratic Trinomial

Solve the quadratic equation.

Case 1) x - 5 = 0
Then x = 5.
This is the answer for case 1.

Case 2) x + 8 = 0
Then x = -8.

x is a segment.
So x cannot be minus.
So x = -8 cannot be the answer.

Case 1) x = 5
Case 2) No roots.

So x = 5.

So x = 5 is the answer.