Given: LN ⊥ KM LN = 16 ft m∠K = 25°, m∠M = 55° Find: Radius R
I drew out triangle LKM in a circle with center O and point N on KM. I tried to use law of sines to get LM and got 27.89514873 but I'm not sure what to do with that or what to do next.
Without seeing the real diagram, there are two interpretation
1. KM is a diameter. But that can't be or else m∠L = 90°.
2. KM is a chord. We know m∠L = 100°
Triangle LNM is tight-angled and tan 55° = 16/NM
NM = 16/ tan 55° = 11.20332... (store that in your calculator's memory)
In the same way, find KN
You now have the chord KM
Let the radius be R
The angle subtended by a chord at the centre is twice the angle at the circle.
So angle KOM = 200°
and by the cosine law
R^2 + R^2 - 2(R)(R)cos 200° = KM^2
solve for R
To find the radius R of the circle, we can use the Law of Sines in triangle LKM. However, we need to find the length of LM first.
1. Use the Law of Sines to find the length of LM:
The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
We have angle K equal to 25°, angle M equal to 55°, and the side opposite angle M is LM. The side opposite angle K is LK.
The Law of Sines can be written as:
LM/sin(K) = LK/sin(M)
Substituting the given values:
LM/sin(25°) = LK/sin(55°)
Rearranging the equation to solve for LM:
LM = (LK * sin(25°)) / sin(55°)
LK is the radius R of the circle, so the equation becomes:
LM = (R * sin(25°)) / sin(55°)
2. Substitute the given value for LN:
We know that LN = 16 ft. We can express LN as the sum of LM and MN:
LN = LM + MN
Rearranging the equation to solve for MN:
MN = LN - LM
MN = 16 ft - LM
3. Use the Pythagorean theorem to relate MN and LM:
In the right triangle LMN, we have:
MN^2 + LM^2 = LN^2
Substituting the known values:
(16 ft - LM)^2 + LM^2 = 16^2
Expanding and simplifying the equation:
256 - 32LM + LM^2 + LM^2 = 256
Combining like terms:
2LM^2 - 32LM = 0
Factoring out LM:
2LM(LM - 16) = 0
This implies that LM = 0 (which is not possible) or LM = 16 ft.
Therefore, LM = 16 ft.
4. Substitute the value of LM into the equation obtained in step 1 to find the radius R:
LM = (R * sin(25°)) / sin(55°)
Substitute LM = 16 ft:
16 ft = (R * sin(25°)) / sin(55°)
Rearrange the equation to solve for R:
R = (16 ft * sin(55°)) / sin(25°)
Use a calculator to evaluate:
R ≈ 27.90 ft
Therefore, the approximate radius R of the circle is 27.90 feet.
thanks this was very helpful but i have a question about the method used to find NM and KN for KN, is there a specific name for it for formula? thank you!
thank you oobleck!
r=19.39057726?
Hmm. That's not what I got. How did you get that?
Or, using another approach,
16/KL = sin25°
16/LM = sin55°
KM^2 = KL^2 + LM^2 - 2*KL*LM*cos100°
KM = 45.515
which is the same answer you get using Reiny's method