استراتژی هندسه - قسمت دوم

Geometry Strategies - Part II

We had a video on geometry strategies toward the beginning of this model, and this video sums up all the strategies in geometry so it will remind a little about what I said in the first video. Remember that geometry is primarily visual and it demands visual as well as logical skills. In that earlier lesson on strategy, I recommended that you always draw a diagram and it, even if one is given on the computer screen, redraw it, so that you can mark it up with what you know and what you can deduce.

It may help to extend a line, or to introduce variables for lengths or angles. Remember the discussion in that earlier video of look big versus look small. In other words details of a relevant geometric relationship may be right next to each other in the diagram or they may be on opposite sides of a complicated diagram. Doesn’t matter whether the triangle is tiny or big, the three angles and it still add up to 180.

So you need to be able to train your eye to see both the small shapes, as well as the large shapes that cross over many others. Wherever there are more than two triangles, especially if parallel lines of some kind are involved, then look for similar triangles. Remember how widely applicable the Pythagorean theorem is. Any time there is a right angle, it’s worth asking yourself if the Pythagorean theorem applies.

In particular, many problems about finding the length of an altitude or the length of a diagonal are Pythagorean problems in disguise. For example, those in the following parallelogram we are given lengths b, y, and x. Well certainly from y and x we could find h. That will allow us to find the area.

And then of course, once we had h we could calculate b minus x if we knew b and x, we could calculate b minus x, and then we could use that to find the length of d. So we could find the area, we could find the, the altitude and we could find the length of the diagonal. In fact, we could even find the length of the other diagonal just by drawing a similar triangle on length c d.

The, the triangle AB, the little right triangle with AB as a hypotenuse. We could stick one of those on the other end, and then use the Pythagorean theorem again. Of course, know the Pythagorean triplets, and recognize their multiples. Remember that the two special right triangles, the 30-60-90 and the 45-45-90, can be specified with very little information, but give a great deal of information.

That’s exactly why the test likes them. They can just throw out one little hint. Then you know you have the triangle and then you get a ton of information. Remember that every line that crosses a pair of parallel lines creates equal angles in different places. Remember all the rules about angles in circles.

There are several rules there that are important to remember. In particular, remember the two times that right angles can be specified because of circular property. An angle inscribed in a semicircle, and a tangent meeting a radius at a point, at the point of tangency. Now in both of these diagrams, I drew the little perpendicular square, the test would not have to do that.

In other words, the test could just expect you to recognize that it has to be a right angle purely because of the circular properties. My final piece of advice may seem odd. Look around at your world. Look at things. Look at architectural features and designs on product packaging.

Look for parallel and perpendicular lines. Right angles, circles, and other geometric figures. Practice seeing some new geometric figure each day in the world around you. This will train your brain to look for geometric patterns. And these are precisely the visual skills that you need to be successful with geometry.