Geometry: Solving the Area of the Inscribed Triangle
Vložit
- čas přidán 5. 05. 2024
- In this fascinating geometry tutorial, delve into the intricacies of calculating the area of an inscribed triangle within a rectangle. Join us as we unlock the secrets behind this geometric puzzle, exploring step-by-step methods to determine the precise area of the triangle nestled perfectly within the confines of the rectangle. Whether you're a math enthusiast seeking to deepen your understanding or a student preparing for exams, this video provides clear explanations and practical techniques to master this geometric concept.
Visit our channel 👉 / @thephantomofthemath
Thats a fun one. Love geometry problems that dont solely rely on the Pythagoras' a^2+b^2=c^3
Let A = area of Δ with one side being the Width of the rectangle,
and B = area of Δ with one side being the Height of the rectangle,
& C = Area (3rd Outer Δ). Then area (Inner Δ) = √{(A+B+C)² - 4AB} =
√{(9+5+4)² - 4(9)(5)} = √{4(9)² - 4(45)} = √{4(36)} = √4.√36 = 2(6) =12.
Love the animation.
Thank you friend!
Excellent problem
I spent far too long trying to solve for a*b like you did, but didn't hit on the obvious way you did it. I think the problem was that I was labeling the sides of the "4" triangle as x and y, which does not produce nice expressions. I gave up for a while, and as I was trying to sleep last night, I came up with an entirely different solution. Let's see if I can intelligibly describe it in pure text...
While I was flailing around with your solution, I had drawn some lines breaking the rectangle into quadrants at the corners of the "4" triangle. I'll label the quadrants
A B
C D
I had this random thought: If I know the areas of A, C, and D, can I find B? So I derived what is probably a well-known equality that A*D = B*C.
So, "4" fills up half of C so C = 8. "9" fills up half of the top two, so A + B = 18, or A = 18 - B. "5" fills up half of the right two, so B + D = 10, or D = 10 - B
Plugging these into the equality, we get (18 - B) * (10 - B) = B * 8 => 180 - 28*B + B^2 = 8*B => B^2 - 36*B + 180 = 0 which is the same quadratic equation you ended up with. In this case, the appropriate solution is "6", so A = 18 - 6 = 12, and D = 10 - 6 = 4, and the total area of the rectangle is A+B+C+D = 30. At this point, the last step is identical, and I got the answer of 12.
Great job, man! Really nice! I love it when people come up with their own solutions and approaches to the problem! Nice! 👍
@ThePhantomoftheMath
I agree! Half the reason why i go to math channels is to see how people solve these kind of problems.
Its like comparing tool kits.
Phantom😊
Bakıdan salamlar.Əla həll etdiniz.Təşəkkürlər.
Needed Google Translate for this but: Çox sağ ol dostum!
I came up with 10
If that's no bother, next time, could you draw something that bears some kind of resemblance with reality?
Not that it made any difference in this specific case.
Actually, the incoherence between shape and numbers was so outrageous and obvious that I immediately redraw the shape according to the numbers. It took me less than 3 minutes from there on.
You could say that your obvious attempt at making this very easy problem harder than it was made it even easier. Or maybe that was your plan all along.
Hi. Thank you for your feedback. I appreciate you taking the time to watch the video and share your thoughts.
I apologize if the visual representation didn’t match the numbers as clearly as it should have. My intention is always to make the content helpful and accurate, and I’m glad to hear you were able to work through it quickly on your own.
Just a quick note: the sketches in math videos are never to scale; they’re meant to illustrate concepts rather than provide precise measurements. However, I’ll make sure to be more careful in future videos to ensure the shapes and numbers align better.
Thanks again for pointing this out, and I hope you continue to find value in the channel. All the best!
@ThePhantomoftheMath Actually, thinking about it, maybe the best formula for learning AND remembering would be to describe the shape clearly without actually showing it, and listing the available data, then let the viewer draw a shape accordingly and which is coherent with the data provided.
That would prevent the viewer to be tempted to find a quick shortcut looking at the shape for clues (if this one has the right proportions) and, at the same time, prevent any misleading proportions to lead to huge waste of time.
It's just an idea. The visual in the thumbnail would be lacking, but the essence of the problem can be described briefly in a colorful snapshot, with attractive words and symbols.
@@kextrz I'll definitely take your idea into consideration for future videos. Balancing clarity, engagement, and educational value is always a priority, so exploring different presentation styles could be beneficial.