Chord Length: A Mathematical Observation

In summary, the conversation is about finding the equation for chord length using the given information of arc length and radius. The first step is to calculate the angle $\theta$ from the arc length formula, and then use the law of cosines and double-angle identity for cosine to find the chord length. It is mentioned that the SAT question only requires knowing which expression to use, and that the given angle of 1.7 is not the one needed for the calculation, indicating that the answer is option D.
  • #1
karush
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View attachment 8904

Ok this should be just an observation solution ..
But isn't the equation for chord length
$$2r\sin{\frac{\theta}{2}}=
\textit{chord length}$$

Don't see any of the options
Derived from that..
 

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  • #2
Hi karush.

All the information is there. You just need to calculate the angle $\theta$ from the arc length $s=r\theta$.
 
  • #3
Olinguito said:
Hi karush.

All the information is there. You just need to calculate the angle $\theta$ from the arc length $s=r\theta$.

It is asking for chord lenght!
 
  • #4
I would start with the arc-length formula to find the subtended angle:

\(\displaystyle \theta=\frac{s}{r}=\frac{3.4}{2}=1.7\)

Then, use the law of cosines:

\(\displaystyle \overline{DF}=\sqrt{2^2+2^2-2(2)(2)\cos(1.7)}=2\sqrt{2-2\cos(1.7)}\)

Lastly, a double-angle identity for cosine:

\(\displaystyle \overline{DF}=2\sqrt{4\sin^2(0.85)}=4\sin(0.85)\)
 
  • #5
Ok
So that's where .85 comes from
So then it's D
 
  • #6
karush said:
It is asking for chord lenght!

And to do that, you need to know the angle $\theta$, don’t you? Calculate that first!
 
  • #7
Why of course we do!

However for this SAT question
It is only asking which
Expression to use
We should know that 1.7 is not the $\theta$ we need
and we have use sin $\theta$
So even without any calculations we should see that it is D
 
Last edited:
  • #8
You were asking
karush said:
Don't see any of the options
That was because you were given arc length $s=3.4$ (and radius $r=2$) but not $\theta$. I was therefore instructing you to compute $\theta$ from the formula $s=r\theta$ so you could use it in the formula $2r\sin\dfrac{\theta}2$ for the chord length.
 

FAQ: Chord Length: A Mathematical Observation

What is chord length?

Chord length is the straight line distance between two points on a circle's circumference, passing through the interior of the circle.

How is chord length calculated?

The chord length can be calculated using the formula c = 2r sin(a/2), where c is the chord length, r is the radius of the circle, and a is the central angle in radians.

What is the significance of chord length in mathematics?

Chord length is an important concept in geometry and trigonometry, as it is used to calculate the lengths of segments and arcs in circles and other curved shapes.

Can chord length be used in real-world applications?

Yes, chord length is used in various fields such as engineering, architecture, and navigation to calculate distances and angles in circular structures and objects.

How does chord length relate to other mathematical concepts?

Chord length is closely related to other geometric and trigonometric concepts, such as arc length, central angle, and radian measure. It is also used in the Pythagorean theorem to calculate the length of the hypotenuse in a right triangle.

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