Hard Leaving Cert Geometry!

ray giraffe

Q. Prove that angle DHC = angle DEC.


From Leaving Cert Project Maths sample paper.


The only proof I know is:

"DHEC is a cyclic quadrilateral since opposite angles sum to 180.

Then DHC and DEC are equal since they both stand on the arc DC."


Leaving Certs need to know that a cyclic quadrilateral has opposite angles summing to 180. However the converse (which the above proof uses) is not on their course.

There must be some other proof, perhaps using similar triangles?

moycullen14

It is a tricky one.

You would HAVE to know that a cyclic quadrilateral can have a circle circumscribed (ie all four points on the quad CDHE lie on the one circle. Once you have that, the proof is trivial.

Actually, the more I think of it, the question is badly worded. It should be Show rather than prove otherwise the question, to my mind, requires the student to prove that angles subtended from the same chord on a circle are equal.

Without a circle, the problem becomes very difficult - at least for a LC student.

ray giraffe

I was talking to another teacher about this, turns out that constructing the circle around CDHE is the intended solution.

It's odd because I hadn't seen any mention of recognising such quadrilaterals in any textbooks.

derb12

Nice question. I would think that recognising the cyclic quadrilateral is fair game on a LC HL question.

MathsManiac

ray giraffe wrote: »

...
Leaving Certs need to know that a cyclic quadrilateral has opposite angles summing to 180. However the converse (which the above proof uses) is not on their course.
...

My interpretatation of the syllabus is that knowledge of whether or not the converses of the various theorems and propositions are true or not, and the ability to apply such knowledge, is certainly intended as part of the syllabus. And this converse is listed in the course document as a "remark".

Also, you could come at it from the perspective of the other corollary to the angle-at-the-circle theorem: the angle in a semicircle is a right angle. The converse of that tells you that if PQR is a right angle, then the circle whose diameter is I]PR[/I passes through Q. In this case, it's clear then that a circle constructed on diameter I]HC[/I will pass through D and E.

moycullen14 wrote: »

...
Actually, the more I think of it, the question is badly worded. It should be Show rather than prove otherwise the question, to my mind, requires the student to prove that angles subtended from the same chord on a circle are equal.
...

I disagree. "Show" and "Prove" are basically synonymous in this context; (there's perhaps a slight implication that you'll get away with slightly less rigour with "show", but I wouldn't rely on it).

There's a well established principle in the LC examinations that, if you're asked to prove one of the specified theorems on the geometry course, you're allowed to assume any of the ones listed before it, along with all the axioms, while if you're asked to prove an "unseen" result that is not one of the listed results, then you are allowed to use any of the listed results on the geometry course. This is a reasonable principle, in my view, since otherwise you could argue that everything would have to be proven from the axioms every time.

ray giraffe

MathsManiac wrote: »

My interpretatation of the syllabus is that knowledge of whether or not the converses of the various theorems and propositions are true or not, and the ability to apply such knowledge, is certainly intended as part of the syllabus.

Pie in the sky. It's easy to make up questions that don't use any maths outside the Leaving Cert syllabus but which would not be answered correctly by any students. e.g. Olympiad questions.

You should work for the department of education. They have told enquiring teachers that the result of the central limit theorem is effectively on the leaving cert course for 2012 (even though it's not on the syllabus) since "simulation is on the course". Rubbish.

The sample papers are hopelessly difficult. 95% of maths teachers are not able to solve many of these questions before reading the solutions.

Most maths graduates would not see the trick needed to solve this question, despite quickly understanding the solution.

Turtwig

I'm actually pleasantly optimistic for Project Maths now.

ChemHickey

When we did it, we said that if you connect a line from H to C. and then construct a semi circle around the points ( I can't remember which ones I think it was H,C,B, but we used different labels and the diagram is WAAY too large.. but it ended up being that the two bases lay on the same arc and therefore are equal