Circle Geometry Reagan Choi Basic Formulas A circle

Circle Geometry Reagan Choi

Basic Formulas A circle is the set of points that are equidistant from one point. The distance is called the radius; twice the radius is called the diameter. The circumference of a circle is 2πr. The area of a circle is πr 2.

Basic Formulas cont. -Pythagorean Theorem: a 2 + b 2 = c 2 -Area of a rectangle and triangle -45 -45 -90, 30 -60 -90 right triangles -Also: similar triangles, and other angle formulas

Problems There are several (difficult!) problems that only require the facts/formulas mentioned so far. Here a few of them: A house is a rectangle that is 60 feet wide and 80 feet long. Lance wishes to run a lap around the exterior of the house, making sure that he is always at least 10 feet away from the walls of the house. What is the shortest distance he can run? A doghouse is a regular hexagon with side length of 4 meters. The dog is tied by a 6 -meter leash to one of the corners of the doghouse, and the leash cannot stretch. What is the area of land that the dog can go to?

Problems In the diagram shown, COED is a square. The radius of circle O is 6 inches. What is the length of AC, in inches? A semicircle of radius 5 is on top of two semicircles with the same radius. What is the area of the shaded region?

Problems Find the area of the annulus (green region) in terms of R and r. Then, find the area in terms of d. Martin the Martian is in a perfectly circular crater on Mars. Martin can walk east 6 meters, west 8 meters, or north 12 meters, to be on the perimeter of the crater. What is the area of the crater?

Angle Formulas Angle BAC = Angle BDC. Angle BAD + Angle BCD = 180. Angle A = (100 -30)/2 ^^^ ← x = (32+54)/2 ← x=y

Problems In the figure below, <R=36 degrees, and <T=42. Find the number of degrees in <RQV. In the figure shown, AB=BC=AC=CD=10, AD=13. What is <ADB?

Length Formulas Power of a Point: For ANY point P (inside or outside the circle), if A, B, C, D lie on the circle such that PAD and PBC are collinear, PA*PD=PB*PC. Ptolemy’s Theorem: AB*CD + BC*DA = AC*BD

Problems Points A, B, C, and D lie on a circle. If AB=8, AP=2, and PC=4, determine the ratio of the area of quadrilateral PAEC to the area of triangle ABE. Let ABCD be a cyclic quadrilateral where AB=4, BC=11, CD=8, and DA=5. If BC and DA intersect at X, find the area of XAB.

Problems In the figure, arc AFD has its center at C, arc AED has its center at B, AD = DB = 20 and m∠ACD = 2(m∠ABD). What is the area of the shaded region between the two arcs? Let O be the circumcircle of isosceles triangle ABC with AB=BC=17. Let the tangents at A and B intersect at D. Line CD meets circle O again at point E. Let F be the intersection Of BD and AE. If AD=24, find BF.

Problems In the figure shown below, circle B is tangent to circle A at X, circle C is tangent to circle A at Y, and circles B and C are tangent to each other. If AB = 6, AC = 5, and BC = 9, what is AX? Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the semicircle from C intersects side AD at E. What is the length of CE?

Homework 1. A circle of radius 1 is surrounded by 4 circles of radius r. What is r? (bottom left diagram) 2. Two circles of radius 5 are externally tangent to each other, and internally tangent to a circle of radius 13 at points A and B, as shown. Find AB. (bottom right diagram)

Homework 3. Circles A, B, and C each have radius 1. Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside circle A and circle B? 4. A circle of radius 1 is tangent to a circle of radius 2. The sides of triangle ABC are tangent to the circles as shown, and AB=AC. What is the area of triangle ABC?

Homework 5. Cyclic quadrilateral ABCD has an incircle. Its sides are AB = 130, BC = 110, CD = 70, and DA = 90. The point of tangency of the incircle to AB divides AB into segments of lengths x and y. What is the positive difference of x and y? 6. Let P be a point on arc BC of the circumcircle of equilateral triangle ABC such that PB = 3 and PC = 5. AP intersects BC at Q. What is the length of PQ?