A Circle problem all students will face! Geometry students, SAT/ACT, GED, Teacher Certification, …
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- Опубликовано: 6 авг 2024
- How to find the length of a segment - intersecting inscribed chords circles.
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got it 2.5 the 2 parts of intersected segements, when multiplied, are =. thanks for pushing my memory 55 years. scary. great one. it's fun being an old math geek.
I wonder who first discovered that theorem. Probably Euclid. No doubt another "Eureka!" moment. I would have loved being in the room when that happened.
Einstein.
@@terry_willisI doubt it. If you read some of Euclid's _Elements_ you will see that this kind of reasoning was well within his grasp. There were several other ancient Greeks who would also be likely candidates.
i like this... and will think on it for a bit!
generally... for 1 unknown requires 1 equation
so rather than watch the video I googled the problem to find
. . . . . . . .
when two chords of a circle intersect within the circle, the product of the segments of one chord is equal to the product of the segments of the other chord.
is called --->
Intersecting Chords Theorem
÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷
The magic of proof comes by connecting the ends of the two lines together forming two SIMILAR triangles: Being similar means the ratios between corresponding sides are also equal.
..There is only one solution
those are length inscribe in a circle
if we have 4 segments x=5/2
AE=x+5, EC=x-1, BE=x, ED=x+2 [ all of this are lengths ]
for example BE * ED = x² + 2x [ wich suggest x can be -5/2 or 5/2, but the length of the segments cant have a negative length ]
if we incribe this in a coordinate sytem with [0,0] at the intersection point, we can use negative values to calculate x, but regardless the length of every segment is the same a segment going from -5 to 0 has the length 5, the absolute value is 5
..and in this example its also a x² term on each side in the equation wich cancel eachother (it does not have to be so, we have to consider what we are doing)
BE * ED = AE * EC ->
x² + 2x = x² +4x -5 [ x² = x² and can be canceled out from each side ]
2x = 4x -5
-4x + 2x = -5
-2x = -5
x = -5 / -2 [ -5 / -2 = 5 / 2 = 2.5]
x = 5 / 2
Note: -2x = -2(x) = 2(-x) ; -a(-b) = ab ; -a/-b = a/b
Thank you
I absolutely don't remember learning this, even though i had math through college differential equations.
Hi John & everyone struggling with math, Have you ever presented a similar problem to find the diameter of the circle?
Let x = the perpendicular from center to the chord AC and y = perpendicular to chord BD, & let r = radius
Then r^2 = x^2+4.5^2 and also r^2 = y^2+3.5^2 ==> and so y^2 - x^2 = 8 (hyperbola quadratic)
If x = 0, y^2 = 8 & y = +/-Sqrt(8). But if y = 0, x^2 = -8 (not real),
Therefore, r = Sqrt(y^2+3.5^2) = Sqrt(8+12.25) = Sqrt(20.25) = 4.5
Since x = 0, then the chord passes through the center of the circle.
Well, if looks like the longer chord AC is a diameter = 9.
IMO he makes more out of this than it's worth. He could have simply said you need to know a theorem, the "segments of chords theorem", to solve it. Simple theorem and simple to solve. (It's no more difficult than area of rectangle = length X width, e.g.) Just set the product of each individual chords' segment lengths = each other and solve for x.
Where does it state that the 2 lines AC and BD are equal lengths
The problem is to know and understand the formula: X x U = Y x T. But why is that true? One has to prove that before going on. Once you have the formula it is peanuts to get the value of X: that is simple algebra (a linear equation). But the big question remains and that is: Why is x.u = y.t. ?
B->D is not the same length as A->C so x * u != y * t ---at least according to the shown diagram, which confused the hell out of me
Now you know you are certain to face this question, just remember the answer is 2.5. (And for a change many will get right.)
😄
Drawing is not to scale, which might confuse some students. Cord AC = 9; cord BD = 7, but they look almost the same.
And there is no way that BE is longer than CE.
This makes me wonder if the values given are even possible.
Now that I look at it some more, the answer indicates that AE is 3x the length of BE. That's nowhere near what the diagram shows. I've tried to picture what this would look like if it was drawn correctly. I can't do it.
Anyway, I did learn something. If I was ever taught about that theorem, I totally forgot it. Glad to learn (or re-learn) it.
x=5/2
X= 2.5
It should be shown to scale, this is ridiculous.
postulates also called axioms are statements accepted as true. rt ang = 90 so all rt angs are =.
theorems are proven statements. ptyh thoer. see, old math geek.