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.
To find the measure of arc AB, we can use the property that the measure of an arc is equal to twice the measure of the corresponding central angle.
Since the chord with a length of 20 is the diameter, it divides the circle into two equal halves, and its central angle is a straight angle measuring 180 degrees.
Thus, the measure of arc AB is twice the measure of the central angle formed by chord AB.
To find the measure of this angle, we need to find the lengths of the chords 16 and 12. Since these chords are parallel, they create a trapezoid within the circle.
To find the length of the chord with a length of 16, we can draw perpendiculars from the endpoints of this chord to the diameter. This creates two right triangles.
Using the Pythagorean theorem, we can find the length of each triangle's hypotenuse. Let's call the length of the hypotenuse of one of the right triangles 'x'. The other leg of the triangle is half the length of the chord (16/2 = 8) and the other leg is the radius of the circle, which is half the length of the diameter (20/2 = 10).
So, using the Pythagorean theorem:
x^2 = (8^2) + (10^2)
x^2 = 64 + 100
x^2 = 164
x ≈ 12.81
Therefore, the length of the chord with a length of 16 is approximately 12.81.
Similarly, we can find the length of the chord with a length of 12 by using the same method.
Let's call the length of the hypotenuse of one of the right triangles 'y'. The other leg of the triangle is half the length of the chord (12/2 = 6) and the other leg is the radius of the circle, which is half the length of the diameter (20/2 = 10).
Using the Pythagorean theorem:
y^2 = (6^2) + (10^2)
y^2 = 36 + 100
y^2 = 136
y ≈ 11.66
Therefore, the length of the chord with a length of 12 is approximately 11.66.
Now that we have the lengths of the chords, we can find the measure of the central angle formed by chord AB.
Using the law of cosines, we can find the angle formed opposite to the chord AB (which is the angle formed by the chords with lengths 16 and 12).
Let's call this angle 'θ'.
Using the law of cosines:
20^2 = 12.81^2 + 11.66^2 - 2(12.81)(11.66)cos(θ)
400 = 164.3761 + 135.7156 - (299.3572)cos(θ)
400 = 300.0917 - (299.3572)cos(θ)
(299.3572)cos(θ) = 300.0917 - 400
(299.3572)cos(θ) = -99.9083
cos(θ) = (-99.9083)/(299.3572)
cos(θ) ≈ -0.3334
θ ≈ acos(-0.3334)
θ ≈ 109.69 degrees
Now, to find the measure of arc AB, we multiply the measure of central angle θ by 2:
Arc AB ≈ 2 * 109.69
Arc AB ≈ 219.38 degrees
Therefore, the measure of arc AB is approximately 219.38 degrees.
Well, it seems like we're looking for the measure of arc AB in circle p, right? So, let's dive right in!
We know that the chord with a length of 20 is the diameter. Now, since a diameter divides a circle into two equal halves, the arc AB, which is formed by the endpoints of the chords with lengths 16 and 12, will also be divided into two equal parts.
So, if we can figure out the measure of one-half of arc AB, we can just double it to get the measure of the whole arc. Sound good? Great!
Now, let's look at the lengths of the chords. We have 20, 16, and 12. Notice anything interesting? The lengths of the chords form a nice arithmetic sequence! That means we can assume that the chords are evenly spaced around the circle.
Let's call the distance between the chord with length 20 (the diameter) and the chord with length 16 "x," and the distance between the chord with length 16 and the chord with length 12 "y."
Since the chord with length 20 is the diameter, it divides the circle into two semicircles. And since the chords are evenly spaced, we can conclude that the chord with length 16 divides one of the semicircles into two equal parts, creating two 8-unit arcs.
So, one 8-unit arc is formed between the chord with length 20 and the chord with length 16, and another 8-unit arc is formed between the chord with length 16 and the chord with length 12.
Since arc AB is the sum of these two 8-unit arcs, we can say that arc AB measures 8 + 8 = 16 units.
But remember, this is only half of the whole arc AB! So, when we double it, we get the measure of the whole arc AB.
Therefore, arc AB measures 2 * 16 = 32 units.
Voila! The measure of arc AB is 32 units. I hope that puts a smile on your face!
To find the measure of arc AB, we need to determine the central angle that corresponds to arc AB.
Since the chord with a length of 20 is the diameter, it divides the circle into two equal halves. Therefore, the central angle corresponding to this chord is 180 degrees.
Now, let's consider the chord with a length of 16. Since it is below the diameter towards the bottom of the circle, it forms an isosceles triangle with the diameter. In an isosceles triangle, the angles opposite to the equal sides are equal. Therefore, the angle formed by this chord is also 180 degrees.
Similarly, the chord with a length of 12 also forms an isosceles triangle with the diameter, and its corresponding central angle is also 180 degrees.
Since arc AB is made up of the endpoints of the chords with lengths 16 and 12, the measure of arc AB is equal to the sum of the corresponding central angles. Thus,
arc AB = 180 degrees + 180 degrees = 360 degrees.
Therefore, the measure of arc AB is 360 degrees.
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