perplexus.info :: Geometry : No trigonometry, no algebra

No trigonometry, no algebra (Posted on 2005-12-23)

Look at the drawing below, where AB = BC = CD = DT = h, and TD perpendicular to AD:

                                   
                                   o T
                                   |
                                   | (h) 
                                   |
                                   |
  +----------+----------+----------o 
  A    (h)   B    (h)   C    (h)   D

The line AT makes an angle x with AD; the line BT makes an angle y with BD; the line CT makes an angle z with CD.

Using only geometry, prove that (angle x) + (angle y) = (angle z).

Submitted by pcbouhid

Rating: 3.0000 (1 votes)

Solution: (Hide)

There are many ways. One of them:
T o----------o----------o----------o----------o | | | | | | | | | | | | | | | | | | | | o----------o----------o----------o----------o | A| B| C| D| | | | | | | | | | | | | | | | Eo----------o----------o----------o----------o | | | | | | | | | | | | | | | | | | | | o----------o----------o----------o----------o F
Triangle ADT is clearly similar to triangle TFE (rotated by 45 degrees and scaled by sqrt(2)). Thus angle ETF is equal to angle TAD = x, so BCT is 180^o - (x + y), but it's also (180^o - z), so z = x + y.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-06-22 01:56:31
re(2): Complex solution	goFish	2005-12-24 15:16:53
re: Complex solution	MindRod	2005-12-24 12:29:24
re: Solution	MindRod	2005-12-24 12:20:19
Complex solution	goFish	2005-12-24 04:44:58
Solution	Bractals	2005-12-24 04:35:23
With Trig. Identity	Bractals	2005-12-24 03:41:34
a start, maybe	MindRod	2005-12-23 22:24:58
re(2): only geometry?	pcbouhid	2005-12-23 15:09:48
re: only geometry?	Richard	2005-12-23 13:47:41
only geometry?	MindRod	2005-12-23 12:51:35

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