I just ran across a great website containing "The world's hardest easy geometry problems"; these are GREAT - they'll keep you busy for hours!
I know, I know, I still have to post the "official results" for the three "brain bogglers" I wrote about a couple of weeks ago. I will get round to it (honestly), but there are so many interesting things to see and do and so little time to do them all in.
For example ... I was just introduced to an absolutely fantastic website: Keith's Think Zone, which – in Keith's words – contains: "Fun and interesting miscellany for school kids of all ages (teachers and other adults are also permitted)".
You could lose yourself for hours in this site. But the problem that brought me here in the first place is a real corker! In fact there are two problems as follows:
The world's second-hardest easy geometry problem
Consider the following diagram (note that this is NOT to scale):
Now, using only elementary geometry (see notes below), determine the value of the angle x. Provide a step-by-step proof.
The world's hardest easy geometry problem
This is very similar to the problem above, but subtly different. Consider the following diagram (once again, note that this is NOT to scale):
As before, using only elementary geometry (see notes below), determine the value of the angle x. Provide a step-by-step proof.
You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees. You may not use more advanced trigonometry, such as the laws of sines and cosines, etc.
You can't go out on the web to find someone else's solution (that would be cheating). You simply have to sit down and grind your way through this yourself. Also, don't tell me the answers – I want to work these out for myself. (Please feel free to email me telling me how you get on, just don't tell me how you did it.)
If you come up with a solution or have any questions, you can post a topic in the "Forums" section of the Programmable Logic DesignLine site for other readers to peruse, ponder, and respond (I promise that I won't look at this particular topic).
The following summarizes everything you need to know to solve the above problems:
Lines and Angles
- When two lines intersect, opposite angles are equal and the sum of adjacent angles is 180 degrees.
- When two parallel lines are intersected by a third line, the corresponding angles of the two intersections are equal.
- The sum of the interior angles of a triangle is 180 degrees.
- An isosceles triangle has two equal sides and the two angles opposite those sides are equal.
- An equilateral triangle has all sides equal and all angles equal.
- A right triangle has one angle equal to 90 degrees.
- Two triangles are called similar if they have the same angles (same shape).
- Angle-Angle (AA): Two triangles are similar if a pair of corresponding angles are equal.
- Two triangles are called congruent if they have the same angles and the same sides (same shape and size).
- Side-Angle-Side (SAS): Two triangles are congruent if a pair of corresponding sides and the included angle are equal.
- Side-Side-Side (SSS): Two triangles are congruent if their corresponding sides are equal.
- Angle-Side-Angle (ASA): Two triangles are congruent if a pair of corresponding angles and the included side are equal.
Questions? Comments? Feel free to email me – Clive "Max" Maxfield – at firstname.lastname@example.org). And, of course, if you haven't already done so, don't forget to Sign Up for our weekly Programmable Logic DesignLine Newsletter.