Most Geometry Students Miss this One Step!
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- Опубликовано: 12 сен 2024
- Do you remember the step that makes this problem EASY?
✴️Try this inscribed angle problem... • Inscribed Angles - Qui...
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✏️Thoughts? Would you do it differently? Questions? Request for a certain math topic? Leave a comment! I'd love to hear from you!
You don't even need that formula. CDE and the circle centre form a 4-sided polygon, and you know 3 of the angles within. 90, 90, 130, so the last angle must be 50.
Yes. There are multiple ways to solve. But if a geometry teacher presents the theorem about tangents/secants intercepting outside a circle and wants kids to use it, I’ve got it here with user-friendly language.
I like multiple ways to solve a problem, but I’m using this theorem…”If two secants intersect outside a circle, then the measure of the angle formed is equal to half the difference of the measures of the intercepted arcs” because it works for tangents, too. Also, if there are 2 chords intersecting INSIDE the circle, the theorem for that is almost identical except it uses the sum of the intercepted arcs. When I taught this IRL, I would do both types of problems side by side. Kids were pretty good about remembering the theorem because it was similar to previous knowledge - finding the average of two numbers.)They then observed you have to change it to subtraction if the intersection was outside. Now to PROVE the theorem, I’d have to use quadrilaterals. More fun for adults who enjoy math, but not so much for teens who just want to pass a required class and who get a formula sheet on their final exam.😊
Great presentation!
Out of curiousity, how the equation angle = (big - little) / 2 was presented in class.
Thanks!
When I taught it, I did use the phrases “big arc” - “little arc” at first, then we’d shorten it to “big-little” after a while.
Or you can solve it in a much easier, less error prone AND more enlightening way, without having to use highly specific formulas that are hard and annoying to memorize and only serve to make students frustrated with math.
Just draw two lines from the centre of the circle to the two tangent points C and E. Now observe that you have a quadrilateral with three known angles 130°, 90° and 90° so the fourth angle must be 360-130-2*90 = 50 from the sum of angles formula so we get 8x-6=50 and thus x=7.
That seems so obvious to me, I thought the missing step was solving for x and not the angle that contains it...
You can also recognize the symmetry and divide the image into two vertically from D. Now you just have a right angle triangle, one angle being 65 degrees and the other 4x-3 degrees, and you can use the sum of interior angles relationship.
This big little rule is daft as it only applies in this one specific situation they will likely never encounter again, ever, on their mathematical journey other than specific questions written to test their knowledge of this one relation.
I like that more than using the formula, too. I’ll show it to my algebra kids so they have another way to solve it on their geometry exams. Thanks!
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Surely it would be better for them to learn more general relationships, and experience interesting methods of problem solving, than simply memorising formulae that are so specific to almost never be useful on a mathematical journey?
You can get to the same answer just as fast by recognising the symmetry and dividing the shape into two down the vertical dimension, realising the tangent implies a right angle, and using just the sum of interior angles of a triangle. And I would argue this is a much MUCH better thing to teach kids as it then gets them learning to exploit tricks in solutions that are widely applicable, rather than a formula that can be easily misapplied because they have been taught to apply a rule, rather than actually reason.
I agree with you. Only teaching algebra this year, but this conversation makes me miss teaching geometry. Thanks!
Or, you could just sum the 3 given angles of the 4 sides polygon, then subtract the result from 360.