Segments Formed by Two Intersecting Secants

Segments Formed by Two Intersecting Secants

How to solve the segments formed by two intersecting secants problems: formula, example, and its solution.

Formula

[blue]*[dark blue] = [green]*[dark green]. [blue]: Exterior segment from a secant. [dark blue]: The secant including [blue]. [green]: Exterior segment from the other secant, [dark green]: The other secant including [green].

[blue]⋅[dark blue] = [green]⋅[dark green]

[blue]: Exterior segment from a secant
[dark blue]: The secant including [blue]
[green]: Exterior segment from the other secant
[dark green]: The other secant including [green]

Example

Find the value of x.

Four segments are formed
by two intersecting secants.

The segments are [x, 3] and [4, 6].

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

x(x + 3) = x2 + 3x
4 + 6 = 10

4⋅10 = 40

Move 40 to the left side.

Then x2 + 3x - 40 = 0.

Factor the left side.

Factor a quadratic trinomial

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

The constant term is (-).
So the signs of the numbers are different:
one is (+), and the other is (-).

(-1, 40), (-2, 20), and (-4, 10)
are not the right numbers.

[-40] = -5⋅8
-5 + 8 = 3 = [+3]
So -5 and 8 are the right numbers.

Use -5 and +8
to write a factored form:
(x - 5)(x + 8) = 0.

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

1) x - 5 = 0
So x = 5.

2) x + 8 = 0
So x = -8.

But this cannot be the answer
because x should be (+).

Solving a quadratic equation by factoring

So x = 5 is the answer.