Segments Formed by Two Intersecting Secants
How to solve the segments formed by two intersecting secants problems: formula, proof, example, and its solution.
(purple)⋅(dark purple) = (blue)⋅(dark blue)
(purple), (dark purple): segments from a secant
(blue), (dark blue): segments from the other secant
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.
[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.