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## High school geometry

### Course: High school geometry>Unit 8

Lesson 3: Arc length (from degrees)

# Challenge problems: Arc length 1

Solve four challenging problems that ask you to find arc length without directly giving you the arc measure.

## Problem 1

In the figure below, start overline, D, B, end overline and start overline, A, C, end overline are diameters of circle P. The length of start overline, P, B, end overline is 8 units.
A circle that is centered around point P. Points A, B, C, and D while line segments D B and A C are diameters. Line segments A P, P B, C P, and D P are radii of the circle. Segment P B is eight units long. The angle A P D is a sixty degree angle.
What is the length of D, C, start superscript, \frown, end superscript?

## Problem 2

In the figure below, the radius of circle P is 6 units.
A circle that is centered around point P. Points A, B, and C Line segments A P, P B, and C P are radii of the circle that are six units long. Segment P B is eight units long. The angle A P B is a one hundred eleven degree angle. The angle B P C is a one hundred twenty-nine degree angle.
What is the length of A, B, C, start superscript, \frown, end superscript
Either enter an exact answer in terms of pi or use 3, point, 14 for pi and enter your answer as a decimal.​

## Problem 3

In the figure below, start overline, B, C, end overline is a diameter of circle P. The length of start overline, B, P, end overline is 3 units.
A circle that is centered around point P. Points A, B, C, and D while line segments B C is a diameter. Line segments A P, P B, C P, and D P are radii of the circle. Segment P B is three units long. The angle A P D is a thirty-five degree angle. The angle A P B is a ninety degree angle. Major Arc A D is highlighted.
What is the length of A, C, D, start superscript, \frown, end superscript?
Either enter an exact answer in terms of pi or enter your answer as a decimal rounded to the nearest integer.​

## Problem 4

In the figure below, angle, A, P, B, \cong, angle, B, P, C. The length of start overline, P, B, end overline is 4 units.
A circle that is centered around point P. Points A, B, and C Line segments A P, P B, and C P are radii of the circle that are four units long. The angle A P C is a one hundred fifty-two degree angle.
What is the length of B, C, start superscript, \frown, end superscript on circle P?
Either enter an exact answer in terms of pi or use 3, point, 14 for pi and enter your answer as a decimal rounded to the nearest hundredth.​

## Want to join the conversation?

• I entered 17.01 for problem 3 but it was marked wrong . Why?
• If you look closely at question 3, it states to round the answer to the nearest integer. An integer is a whole number. Therefore, 17.01 is not a whole number. 17 is the nearest whole number (integer) You had the solution correct, however did not follow the directions to round to the nearest integer
• How does this all work? I read all the solutions, but I still don't understand... Could someone please explain?
• Lets go through the first problem. We are trying to figure out the length of arc DC. Lets find the circumference first. Circumference= 2pi*r which is also equal to diameter*pi. Therefore the circumference of the circle is 16pi. The arc measure is equal to the central angle that intercepts it. Since line CA is a diameter, it is equal to 180 degrees, so if you add the measures of angles DPA and DPC, it will equal 180 degrees. Since the measure of DPA is 60 degrees, DPC is equal to 120 degrees. Now we set up a proportion to find the part of circumference intercepted by that angle, so Arc length/Circumference = Arc measure/Degrees in a circle. Arc length/16pi = 120/360. We can simplify the fraction to the right and it will be Arc length/16pi = 1/3. Since we need to isolate the arc length, we can multiply both sides by 16pi. After that we get Arc length = 16pi/3. The length of arc DC is equal to 16pi/3.
Lets also go through problem 3. We know that arc AC is equal to 152 degrees. We also know that the other two arcs, AB and BC are congruent to each other. Therefore do 360-152/2. That will be the central angle of BPC, which is 104 degrees. The equation for the arc length is this: Central angle/360 = Arc length/ Circumference. Since the radius is four the circumference will be eight. The equation is 104 / 360 = s/8pi. Multiply both sides by 8 pi since we need to isolate s, and you should end up with the answer which is 104*8pi / 360 = s.
Hope this helps!
• Just to clarify for everybody if you press alt and then the letter P the pi symbol will pop up.
• It did not work on my keyboard, I put in pi and it changed to the symbol or ALT 227 on the number pad on the side with numlock on.
• For number four, I answered with 104/45 times 3.14 and it was marked wrong. Then I checked my answer and the only difference between the answer provided and mine is that they have the pi sign instead of 3.14. I don't think it is fair to mark an answer wrong if you don't provide the sign in the drop down box. I was just wondering if I was wrong?
• Writing pi instead of just 3.14 is correct, because even though the pi sign isn't provided, you just spell out pi rather than 3.14, because pi means 3.1415926...., while just 3.14 is incorrect.
(1 vote)
• With some of these questions, there is no option to use the 'pi' symbol.
• It’s because you have to round or use 3.14 instead of pi.
• 360-152=108 not 104 a small mistake ?
• Hey.. we get 208. After this we divide it by 2 to obtain the angle. Hence, we get the angle as 104 deg