Segments Formed by Two Intersecting Secants

Segments Formed by Two Intersecting Secants

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

Formula

If two secants start from the same point, then the product of the secant's segments is equal to the product of the other secant's segments.

(purple)⋅(dark purple) = (blue)⋅(dark blue)

(purple), (dark purple): segments from a secant
(blue), (dark blue): segments from the other secant

Proof

Segments Formed by Two Intersecting Secants: Proof of the Formula

Draw the green arc
and two additional white chords like above.

The red angles are the inscribed angle
of the green intercepted arc.
So the red angles are congruent.

Draw the red dot angle
at the intersecting point of the secants.

See these two triangles.
These two have two pairs of congruent angles.

So, by the AA similarity,
these two triangles are similar triangles.

So the corresponding sides are proportional:
(ratio of the longest sides) = (ratio of the shortest sides).

Similarity of sides in triangles

Then (purple)⋅(dark purple) = (blue)⋅(dark blue).

Proportion

Example

Find the value of x. Segments formed by two intersecting secants: x, 3, 4, 6.

[x, (x + 3)] and [4, (4 + 6)]

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

Solve the quadratic equation by factoring.
Then x = -8 and 5.

But x ≠ 8.
(∵ x > 0)'∵' means 'because'.

So x = 5.