Can You Solve Without Using Trigonometry?

Added: 2026-05-12

[00:00:00]All right, this one is really beautiful.

[00:00:05]We've got a right triangle A B C.

[00:00:12]The right angle is at B.

[00:00:15]Angle C is 36° and the hypotenuse AC has length 2.

[00:00:25]We need to find the base B C which is X.

[00:00:31]And here's the catch.

[00:00:34]No trigonometry, no sign, no cosine, just pure geometry.

[00:00:44]So we need a different idea.

[00:00:49]Here's the trick. Extend the bottom side to the left past point B to a new point P.

[00:01:04]Now connect A to P and choose P carefully so that angle A P C is also 36°.

[00:01:20]Now look at the big triangle we just created.

[00:01:25]Triangle A P C.

[00:01:30]Angle at P is 36°.

[00:01:34]Angle at C is also 36°.

[00:01:40]So the triangle is isles which means the opposite sides are equal.

[00:01:49]Since AC is two, A is also two. Now look at the two right triangles sitting side by side.

[00:02:03]Triangle A B C and triangle A B P both have a right angle.

[00:02:14]Both share the same height and both have hypotenuse 2. So they're congruent.

[00:02:24]That means the left base PB is equal to the right base B C. And since BC is X, PB is also X. So the full bottom length P C is 2X.

[00:02:46]Keep that in mind. Now comes the next construction.

[00:02:52]Extend the line P A upward to a new point Q.

[00:03:00]Then connect C to Q.

[00:03:05]Choose point Q carefully. So that angle A CQ is 36°.

[00:03:15]Now let's examine the giant triangle P CQ.

[00:03:23]Angle at P is 36°.

[00:03:27]Angle at C is 36 + 36 which is 72°.

[00:03:38]Because all angles in a triangle must add up to 180. That top angle at Q has to be 72° as well.

[00:03:50]That means triangle P CQ isles 2.

[00:03:57]Now look at the smaller triangle on the right.

[00:04:02]AC Q.

[00:04:04]Its top angle is 72.

[00:04:08]Its bottom angle is 36.

[00:04:12]So its third angle right near the middle must also be 72.

[00:04:20]So that triangle is also isosles.

[00:04:25]Since AC is 2, CQ is also two.

[00:04:32]Let's call the small top segment AQ equal to A.

[00:04:40]So the full left side PQ becomes 2 + A.

[00:04:48]And because triangle PCQ is isoclesles, its bottom side PC is also 2 + A.

[00:05:00]Now we use one final idea.

[00:05:05]Similar triangles.

[00:05:09]The large triangle P CQ and the small triangle A C Q have the exact same angles.

[00:05:26]So their side ratios match.

[00:05:30]Set up the proportion.

[00:05:32]2 + a / 2 = 2 over a.

[00:05:42]Cross multiply a * 2 + a = 4.

[00:05:51]Expand it. A 2 + 2 a = 4.

[00:05:58]Move everything to one side.

[00:06:01]a^ 2 + 2 a - 4 = 0.

[00:06:08]Now solve the quadratic.

[00:06:11]Using the quadratic formula, we get a =1 plus or minus<unk> 5.

[00:06:28]Since a is a length, we ignore the negative solution.

[00:06:37]So a equ=<unk> 5 - 1.

[00:06:47]Now go back to PC.

[00:06:51]PC equals 2 + a.

[00:06:58]So that becomes 2 +<unk> 5 - 1 which simplifies to<unk> 5 + 1.

[00:07:13]But PC also equals 2x.

[00:07:19]So 2x =<unk> 5 + 1 divide by 2 and finally x =<unk> 5 + 1 / 2 and yes that is exactly the golden ratio.

[00:07:45]Pretty amazing, honestly.

[00:07:48]A simple 36 degree triangle quietly hiding one of the most famous numbers in mathematics.

[00:07:57]All right, see you in the next one.