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First, draw line PO to obtain triangle OPR. We know that a triangle inscribed a half circle with one side being the diameter of a circle is always a right triangle. This means angle OPR is a right angle. Given that angle PRO is 35, we know that angle POR is 180-90-35 = 55

Set center point of circle = C Since POR = POC = 55, this means OPC = 55, and PCO = 180 - 55 - 55 = 70. Using symmetry, we know that angle PCQ is 180 - 70 - 70 = 40

To find the length of minor arc PQ, we need to get the angle extended by the arc at the center of the circle. Name the center as C. We need to find the angle PCQ and substitute in the formula (θ/360)*2πr where θ is the angle subtended by the arc at the center.

Step 1:

Since the lines OR and PQ are parallel and Angle ORP is 35 degrees, as per alternate angles rule, angle RPQ is 35 degrees.

Step 2:

Draw a line from P to C. Line PC is equal to CR, which is the radius i.e 9. Now consider the triangle RPC. In this triangle, 2 sides are equal PC and CR and the angle CRP is 35 degrees. As per the rules of isosceles triangle, angle RPC must also be 35 degrees.

So, the total angle CPQ is 70 degrees.

Step 3:

Draw a line from Q to C. So, line PC is equal to line QC. If angle CPQ is 70 degrees, then angle PQC will also be 70 degrees.

Finally, angle PCQ is 180-140 = 40 degrees

Step 4:

Substitute 40 in the formula (θ/360)*2πr = (40/360)*(2*π*9) = 2π

In the circle shown, PQ is parallel to diameter OR, and OR has a length of 18. What is the length of the minor arc PQ?
a) 2pi
b) 9pi/4
c) 7pi/2
d) 9pi/2
e) 3pi

Attachments

arc.JPG [ 11.97 KiB | Viewed 2045 times ]

Last edited by pesquadero on 20 May 2006, 20:09, edited 1 time in total.

In the circle shown, PQQ is parallel to diameter OPand OR has a length of 18. What is the length of the minor arc PQ? a) 2pi b) 9pi/4 c) 7pi/2 d) 9pi/2 e) 3pi

In the circle shown, PQQ is parallel to diameter OPand OR has a length of 18. What is the length of the minor arc PQ? a) 2pi b) 9pi/4 c) 7pi/2 d) 9pi/2 e) 3pi

One thing to keep in mind in Arc question is whether the angle that is given is from either the center of the circle or from the edge of the circle (such as this example)

If angle is from the middle of the circle:
THE LENGTH OF ARC = (angle * pi * Radius)/180

If angle is not from the middle of circle: also known as MINOR ARC
(Minor arc always have twice the measure of the inscribed angle)

therefore in this case ARC OP = 2*X, since X = 35. OP=70
PQ || OR so PQ = 180-70-70 = 40

But remember that 40 in reference to the whole circle is 40/360 = 1/9 of the circumference

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