Geometry: Find the Area of the Circle with Intersecting Chords

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The Phantom of the Math

Күн бұрын

Пікірлер: 33

@eggz2313 4 ай бұрын

who's gonna tell him he's finding the area of Super Smash Bros?

@ThePhantomoftheMath 4 ай бұрын

LOL 🤣🤣🤣

@Darisiabgal7573 3 ай бұрын

6*4 = 2*x 24/2 = x = 12 We can put a parallel line on the right side that mirrors the left The internal horizontal segment is 10. We can another parallel line over the center that mirrors the bottom The internal vertical segment is 2. The diagonal is SQRT(100+4) =2 SQRT(26) z+ 2* SQRT(26) + z = diameter, radius is z + SQRT(26) and area is pi (z + SQRT(26))^2 = (z^2 + z* 2*SQRT(26) +26)*pi z * (z + 2*SQRT(26)) = 24 = z^2 + z * 2 *SQRT(26) Therefore area = (24+26) pi = 50pi units^2

@kraya6740 3 ай бұрын

Thats my solution also.

@artandata 4 ай бұрын

I think there is another the method which is graphical: find the intersection of both lines perpendicular to the chords passing by their respective midpoints, giving so the geometric place of the center of the circle. By means of a compass (opened from center tu any of the ends of chords) draw the circumference then shade the area inscribed to find the solution. regards.

@ThePhantomoftheMath 4 ай бұрын

Thank you for your comment! You’re absolutely right about the graphical method. This method is a classic geometric approach and is very effective. Thanks for sharing!

@Ivan-fc9tp4fh4d 3 ай бұрын

Vertical shift O is (6+4)/2 - 4 = 1. Then we have for blue chord and AC: (r-1)(r+1)=7*7, then r^2=50 and S=50*PI.

@thomasleach9417 3 ай бұрын

I never knew about that 2nd method. Fascinating

@KipIngram Ай бұрын

This is trivially easy using the law of sines. For a triangle inscribed in a circle, with chords a, b, and c and opposite angles A, B, C, we have a/sin(A) = b/sin(B) = c/sin(C) = D, where D is the diameter of the circle. So it's easy to see that the angle formed at the red dot is arctan(6/2) + arctan(4/2) = 135 degrees. That chord is of length 10. So, by the above we have D = 10/sin(135) = 10*sqrt(2). So R = 5*sqrt(2), and area = 50*pi. Easy peasy - doesn't really need a 13 minute video.

@ThePhantomoftheMath 29 күн бұрын

@@KipIngram Hi! Thank you for your comment and for sharing your solution. You’re right-the Law of Sines can definitely be applied to this problem, and it’s a great method. 👍 Regarding the length of the video, I see your point, but I wanted to clarify that in those 13 minutes, I aimed to explain three different methods for solving the problem. I believe it’s important to make the content accessible to everyone, especially for those who might not be as proficient in math as you. I always try my best to ensure that the explanations are clear and understandable, and that takes time. Cheers!

@gelbkehlchen 5 күн бұрын

Solution: r = radius of the circle. I place the intersection point of the two chords at the origin of a coordinate system. Then the left end point of the horizontal chord is at A = (-2;0), its right end point at C = (12;0), because the section of this chord results from the Chord theorem in mathematics, 4*6/2 = 12. The end points of the vertical chord are at B = (0;-4) and D = (0;6). The two perpendicular bisectors of these two chords intersect at the center of the circle. The horizontal perpendicular bisector then has the equation y = (6+(-4))/2 = 1 and the vertical perpendicular bisector then has the equation x = ((-2)+12)/2 = 5. The center is then at M = (5;1). According to the Pythagorean theorem, the radius of the circle is: MC² = r² = (12-5)²+(0-1)² = 7²+1 = 50. The area of the circle is then: 50*π ≈ 157.0796

@WunditMarkoMagok 3 ай бұрын

Applying the concepts of a circle passing via 3points,that's a nice job! l learnt this in my school era❤❤ Love the ezplanation

@ThePhantomoftheMath 3 ай бұрын

Thank you. Glad you enjoyed it!

@bkp_s Ай бұрын

Sir, I was known to the formula applied in method two. Thanks a lot sir

@ThePhantomoftheMath Ай бұрын

@@bkp_s Really nice! You are one of the rare ones!

@bkp_s Ай бұрын

@@ThePhantomoftheMath I need to learn classic english too with you in this way

@ThePhantomoftheMath Ай бұрын

@@bkp_s English is not my native language (as you can probably tell 😹), but I'm happy to help if I can.

@bkp_s Ай бұрын

@@ThePhantomoftheMath No need of taking additional bother for this.your vdo and replying any message are sufficient for me. I shall pick it up from there. Thanks once again for taking pain to response me. I am really grateful to you. 🙏

@08-thanutchasamumanit98 2 ай бұрын

I learned math form KZbin

@PrithwirajSen-nj6qq 3 ай бұрын

Points on circles are (0,6), (-2 0) ,(0 -4) Centre (-g, -f) Circle and equation x ^2+y^2+2gx +2fy +c=0 Putting the points in it 36+12f+c =0--(1) 4 - 4 g +c =0- (2) 16-8f +c =.0--(3) From 1 and 3 f = -1 from 3 c=-24 From 2 g = 5 (-g,-f ) =(-5,1) Distance between (radius) (0 ,6).& (-5 ,1) =√(25+25)=√50 Area of circle =50π sq unit. Comment please.

@ThePhantomoftheMath 3 ай бұрын

Yes! Everything checks out! The center is (5, 1), and the area of the circle is 50π square units. Great job!!! Nice take!

@timmeeyh6523 3 ай бұрын

i found that points p1=(x,y), p2 = (x+2,y+6) and p3 = (x+2,y-4) all satisfy r = sq_root(x^2 + y^2) , and was able to solve it as a system of equations 0 = 4x + 4 +12y + 36 && 0 = 4x + 4 -8y + 16 -> x = -3y -10 && x = 2y -5 ---> y = -1 && x=-7 so p1 is at coords y = -1 x = -7 where 0,0 is the center of the circle. This gives a radius equal to sq((-1)+(-7)), and we can just square it and multiply with pi for 50pi I feel intuitively that the horizontal bit with unknown length should be 12, due to shape similarity with triangles on left hand side. I guess it holds, as one can calculate a vector (7,1) from p1 to center of circle given that the sublengths of the chords are known.

@ThePhantomoftheMath 3 ай бұрын

Really nice! Well done!

@tomtke7351 3 ай бұрын

A line thru the center of circle and parallel to the "2" line would have equal distances along the "4", "6" lines; i.e. 5 and 5.

@ThePhantomoftheMath 3 ай бұрын

Yes...and it does just that at 3:49

@someonespadre 4 ай бұрын

I did it in my head. Rsquared=7squared plus 1squared =50, area =50pi

@Programmingwithrohith 4 ай бұрын

What's the formula, what's the method?

@someonespadre 4 ай бұрын

@@Programmingwithrohithcrossing chords theorem: 2x=6*4=12. Radius point is 1 above horizontal line and 7 left of the right end on the arc. R^2=1^2+7^2=50 therefore area equals 50Pi.