Practice Problem - Sai Govindu
Directions: Solve for the radius of the smaller circle - Circle R. RS is 25. Circle S has a radius of 20. The chord in between both circles is 24.
Explanation/Work: To solve for the radius of circle R we have to solve for some missing values of circle S and the right triangles located using the properties of circles. This is a multi-step problem. To start out we know the radius of the bigger circle is 20 and half the chord in between both circles is 24/2 or 12. We have a right triangle created through the chord perpendicular bisector, half of the chord itself, and the radius of circle S. We are already given 2 lengths so we can use Pythagorean theorem to solve for the length or distance from center S to the chord. If we know this value and the value that RS is 25, we can then create another right triangles in circle R and solve for the radius.
Explanation/Work: To solve for the radius of circle R we have to solve for some missing values of circle S and the right triangles located using the properties of circles. This is a multi-step problem. To start out we know the radius of the bigger circle is 20 and half the chord in between both circles is 24/2 or 12. We have a right triangle created through the chord perpendicular bisector, half of the chord itself, and the radius of circle S. We are already given 2 lengths so we can use Pythagorean theorem to solve for the length or distance from center S to the chord. If we know this value and the value that RS is 25, we can then create another right triangles in circle R and solve for the radius.
Pythagorean Theorem: x^2 + 12^2 = 20^2
x^2 + 144 = 400
x^2 = 400 - 144
x^2 = 256
x = 16
The segment TS as we solved is 16. Now because we know TS is 16 and RS is 25 we can subtract to create another right triangle.
x^2 + 144 = 400
x^2 = 400 - 144
x^2 = 256
x = 16
The segment TS as we solved is 16. Now because we know TS is 16 and RS is 25 we can subtract to create another right triangle.
We now created another right triangle with RT being 9 and the other leg still 12. We want to solve for the radius and we are given 2 lengths so we can use Pythagorean theorem again to solve.
9^2 + 12^2 = r^2
r^2 = 81 + 144
r^2 = 225
r = 15
We solved for r and got 15. 15 is the radius of the smaller circle or circle R.
9^2 + 12^2 = r^2
r^2 = 81 + 144
r^2 = 225
r = 15
We solved for r and got 15. 15 is the radius of the smaller circle or circle R.
Practice Problem 2 - Sai Govindu
Directions: Solve for all the unknown lengths given the values in the diagram above. There is an inscribed trapezoid with an exterior angle. The exterior angle continues its secant through AC. Find arc measures and the exterior angle measure and angles BEC and AED.
Explanation/Work: To start out, we can use what we are given about the figure to find some missing values. Because angle ACD is 50 degrees, that means arc AD is 120 degrees. Now we know the rest of the circle has 3 equivalent arc lengths so we can do (360 - 120)/3 to find each arc length that has an equivalent chord. So 240/3 = 80. SO we know arcs AB, BC, CD have a measure of 80 degrees. Now using an interior angle theorem from lesson 7, we can find the measures of angles BEC and CED. BEC: (120+80)/2 = 200/2 = 100 BEC = 100 degrees. CED: we can do this 2 ways , 180-100 = 80. or (80+80)/2 = 80
So finding the arc values through circle properties and inscribed angles etc. we were also able to find other values in an inscribed polygon.
So finding the arc values through circle properties and inscribed angles etc. we were also able to find other values in an inscribed polygon.
Practice Problem 1- Amoagh Gopinath
Find all the values.
Explanation: First you would start of by finding v . You would find v by using the other arcs in the semicircle. You would subtract 70 and 10 from 180 and should get 90 degrees. Next you will solve for w using the same method, but on the other semicircle. You would subtract 80 from 180 to get 100 degrees as the value for w. Next we have to find x. For x you would start of by finding the value of the angle next to it by using the inscribed angles theorem. You would get 5 degrees as the value of that angle. Since the chords form a triangle inscribed in a semicircle x and the neighboring angle together would equal 90 degrees. To find x you would subtract 5 from 90 degrees to get 85 degrees as the value for x. Next you must find y by dividing the value of the inscribed arc by 2. That would mean y is 35 degrees. Finally to find z you would use the cyclic quadrilateral theorem. x and z are supplementary so z is 180-85 which is equal to 95. So z is 95 degrees.
Explanation: First you would start of by finding v . You would find v by using the other arcs in the semicircle. You would subtract 70 and 10 from 180 and should get 90 degrees. Next you will solve for w using the same method, but on the other semicircle. You would subtract 80 from 180 to get 100 degrees as the value for w. Next we have to find x. For x you would start of by finding the value of the angle next to it by using the inscribed angles theorem. You would get 5 degrees as the value of that angle. Since the chords form a triangle inscribed in a semicircle x and the neighboring angle together would equal 90 degrees. To find x you would subtract 5 from 90 degrees to get 85 degrees as the value for x. Next you must find y by dividing the value of the inscribed arc by 2. That would mean y is 35 degrees. Finally to find z you would use the cyclic quadrilateral theorem. x and z are supplementary so z is 180-85 which is equal to 95. So z is 95 degrees.
Practice Problem 2- Amoagh Gopinath
There is a regular pentagon inscribed into a circle. Find out the value of x.
You start by finding the measure of each individual arc. Since a pentagon has 5 sides, you would do 360/5 which is equal to 72. To find x you would have to find half of the difference of the two intercepted arcs. Before that we'll need to find the value of the larger arcs which is two smaller arcs so we would do 72x2 which is equal to 144. Then you would find the difference of 144 and 72 which is 72 and divide it by 2 which is 36. x is equal to 36 degrees.
You start by finding the measure of each individual arc. Since a pentagon has 5 sides, you would do 360/5 which is equal to 72. To find x you would have to find half of the difference of the two intercepted arcs. Before that we'll need to find the value of the larger arcs which is two smaller arcs so we would do 72x2 which is equal to 144. Then you would find the difference of 144 and 72 which is 72 and divide it by 2 which is 36. x is equal to 36 degrees.
Practice Problems: Brice Chen
#1
A poorly designed circular ice rink has a portion under renovation, represented by the shaded region. The central angle of the sector is 120 degrees. How long is the diameter of the ice rink in feet? Leave answers in simplified radical form. a: To solve it, first use the formula for solving for an area of an arc. Substituting the known values, the area of the sector, and the central angle gives you the equation 72pi=pi(Radius^2)(120/360). Simplifying the equation gives 72=Radius^2/3.. The answer is not a whole number, but a radical. 216=Radius^2. Prime factorization of 216 gives 6√6. The diameter is two radii, so the answer is 12√6 ft.. |