Take a piece of paper and note the following on your diagram. It will be difficult to follow without an annotated diagram.
Label the (horizontal) diameter as CD, with C on the left.
Label the chord 16 units long as PA (P on the left).
Label the chord 12 units long as QB (Q on the left).
Label the centre of the circle as O.
Construct a (vertical) diameter perpendicular to CD. Label the ends as E (on top) and F (at the bottom).
Label the intersection PA and EF as X, and the distance OX=x.
Label the intersection QB and EF as Y, and the distance OY=y.
Label the radius of the circle as r = OD.
Confirm that:
OD=OE=OC=OF=r=10
PX=XA=8
QY=YB=6
We don't know yet the distances x and y.
By the theorem of intersection of chords, we have
PX*XA = FX*XE
8²=(10-x)(10+x)
Solve for x to get 6.
θ1=∠AOD=sin-1(6/10)
Similarly,
QY*YB=(10-y)(10+y)
6²=(10-y)(10+y)
Solve for y to get 8.
θ2=∠BOD=sin-1(6/10)
⇒
φ=∠BOA=θ2-θ1
Arc length of AB
=rφ/2π
=10φ/2π
in circle p below the lengths of the parallel chords are 20,16, and 12. Find measure of arc AB..... the chord with a length of 20 is the diameter. the chords with lengths 16 and 12 are below the diameter torwards the bottom of the circle. arc AB is the arc made up of the endpoints of the chords with lengths 16 and 12.
2 answers
In order to find arc AB you need to first find the length of P to the midpoint of B.
For now, you can assume the midpoint of B is C.
You can draw a right triangle of CPB.
PB = 10 because it is the radius, which is half of the diameter (P = 20)
If you then add this to the Pythagorean Theorem, you find that C is 8.
10^2 = 6^2+x^2
100 = 36+x^2
64=x^2
8=x
You can now use tan, sin, or cos to find the angle of P, which is around 36.87 degrees.
For now, assume that the point above point A on the line of P is D.
You need to find the angle of the triangle APD. You can assume that it is also 36.87 degrees, since its basically CPB.
mCPD is 90 degrees.
If you take 90 degrees and minus the angles of both triangles (basically 36.87 x 2), then you get the arc of AB, which is 16.26 degrees.
For now, you can assume the midpoint of B is C.
You can draw a right triangle of CPB.
PB = 10 because it is the radius, which is half of the diameter (P = 20)
If you then add this to the Pythagorean Theorem, you find that C is 8.
10^2 = 6^2+x^2
100 = 36+x^2
64=x^2
8=x
You can now use tan, sin, or cos to find the angle of P, which is around 36.87 degrees.
For now, assume that the point above point A on the line of P is D.
You need to find the angle of the triangle APD. You can assume that it is also 36.87 degrees, since its basically CPB.
mCPD is 90 degrees.
If you take 90 degrees and minus the angles of both triangles (basically 36.87 x 2), then you get the arc of AB, which is 16.26 degrees.
1 answer