Solve the length AB (the dashed line)

Question

Can someone show me how I can solve this? (Step by step example with solution appreciated a lot as I am currently practicing).

EDIT: After a closer look, it looks as if this is an Isosceles Triangle (http://mathworld.wolfram.com/IsoscelesTriangle.html).

P.s. I appreciate if someone could share with me the mental of thinking when something similar appears, and show me how to solve this one, I'd just like to know since I do not remember much.

EDIT: Can someone copy my photo and mark things out, ABC etc, that would make it far more simple for me to understand and be able to follow. Thanks a lot in advance.

Answer 1

You have been given with the length of the arc(S), Using that and the radius and this formula below, Calculate $\theta$

$S=r\theta$, $\theta$ is the angle at the centre in radians.

So,

$25=10\times\theta$

$\theta=2.5$ radians

Or,

$\theta \approx 143^\circ$ in degrees

Now, In $\triangle OAB$ where O is the centre of the circle, the perpendicular from O will bisect $AB$, because $\triangle OAB$ is an isosceles triangle.

Let's say at C.

So, the overall Length of $$AB=AC+CB=10\times \sin 71.5^\circ+10\times \sin 71.5^\circ=18.96$$

Answer 2

I'd solve this using two facts:

1. The law of cosines

2. The fact that the angle at the center of the circle is the arclength subtended by the two radii, divided by the length of the circle radius.

In this case, the arclength is 25 and the radius length is 10, so the angle is 2.5 radians (which is 180 * 2.5 / pi degrees, if you prefer degrees).

The law of cosines then tells us that the long side has length $s$ where

$$ s^2 = r^2 + r^2 - 2rr \cos b $$ where $b$ is the angle and $r$ is the radius, so $$ s^2 = (10)^2 + (10)^2 - 200 \cos (2.5) \\ = 200 (1 - \cos(2.5)) \approx 18.97 $$ where I worked out this last value with a calculator.

As for "what were my thought processes", it was basically "remembering 10th grade geometry class." The law of cosines comes from the side-angle-side congruence theorem, and the relationship of angle to the arc subtended came from the chapter on circles. Both of these came from "School Mathematics Geometry," a book that was in vogue in 1970, back when I was in 10th grade. So one (lousy) answer for "how do I approach problems like this in the future?" is "Find that book and do about 30% of the exercises in it." To be honest, I probably would not remember most of what was in the book except that I happen to work in computer graphics, and have done math and computer stuff all my life. So I'm probably not a good person to ask "How can I copy your thought processes?" -- they're the thought processes of someone who's thought like a mathematician for 40 years or so.

Answer 3

x degree/360 degree=25/2*pi*10 so x degree=143.18 and then use law of cosine formula