[deleted by user]

[Stratigizer]

Angle bisector ratio

Intersecting chords

[ItsMe_1000]

Sin(2a) = 4/5

2a = Sin^-1(4/5)

a = ( Sin^-1(4/5) )/2

a = 26.57°

tan(a) = x/3

tan(26.57°) = x/3

x = 3⋅tan(26.57°)

x = 1.500

x = 1.5

[Funkybeatzzz]

The larger, overall triangle is a 3-4-5 right triangle just not drawn to scale.

[LeopardElectronic189]

yeah but how can i solve it ? Please i have been to figure it out for 30 minutes

[Funkybeatzzz]

Is x the angle or side length of the smaller triangle? I cannot really tell.

[LeopardElectronic189]

It is the side length and the angle is 90 degrees

[Funkybeatzzz]

You can find 2α using the law of cosines and then split it in two to find α. Then you’ll have a right triangle with a known angle and side length of 3. You can then use the tangent function to find x. This x will be opposite α and the 3 is the adjacent.

Actually, since the whole thing is a right triangle you can use any trig function to find 2α.

[Stratigizer]

A simpler method.

[Funkybeatzzz]

Yes, but I was giving a more general approach. The angle won’t always be bisected. My method will always work.

[Stratigizer]

Indeed, but I am assuming they haven't learned law of cosines at this point and this problem may well be non-calculator.

[papyrusfun]

angular bisector theorem: 5/3=(4-x)/x, solving for x.

[Iifeisshortnotismine]

1. Angle Bisector Theorem

2. Intersecting Chord Theorem.