Geometry Chapter 10 Circles Test

Geometry Chapter 10 Circles Test Geometry Chapter 10 Circles Test Instructions Answer all questions to the best of your ability Show all work for full credit This test covers concepts related to circles including their properties relationships and applications Part 1 Multiple Choice 2 points each 20 points total 1 Which of the following is NOT a property of a circle a All points on the circle are equidistant from the center b The diameter is twice the length of the radius c A circle is a threedimensional shape d The circumference is the distance around the circle 2 What is the name of the line segment connecting two points on a circle and passing through the center a Radius b Diameter c Chord d Tangent 3 A central angle of a circle measures 60 degrees What is the measure of its intercepted arc a 30 degrees b 60 degrees c 120 degrees d 180 degrees 4 Two circles are congruent if they have the same a Radius b Diameter c Circumference d All of the above 5 What is the relationship between an inscribed angle and its intercepted arc a The inscribed angle is twice the measure of its intercepted arc 2 b The intercepted arc is twice the measure of its inscribed angle c The inscribed angle is half the measure of its intercepted arc d The intercepted arc is half the measure of its inscribed angle 6 A tangent line to a circle intersects the circle at a Two points b One point c No points d Infinitely many points 7 The formula for the circumference of a circle is a r b 2r c d d Both b and c 8 The area of a circle with a radius of 5 units is a 10 square units b 25 square units c 50 square units d 100 square units 9 What is the name of a line segment connecting two points on a circle but not passing through the center a Radius b Diameter c Chord d Tangent 10 A secant line to a circle intersects the circle at a Two points b One point c No points d Infinitely many points Part 2 Short Answer 4 points each 20 points total 1 Define the following terms a Radius b Diameter 3 c Chord d Tangent 2 Explain the difference between an inscribed angle and a central angle 3 State the formula for the area of a circle and explain the meaning of each variable 4 Describe the relationship between the measures of two inscribed angles that intercept the same arc 5 What is the measure of a central angle that intercepts a semicircle Explain your reasoning Part 3 Problem Solving 6 points each 30 points total 1 A circle has a radius of 10 cm Find its circumference and area 2 A central angle of a circle measures 120 degrees What is the measure of its intercepted arc 3 Two chords intersect inside a circle One chord is 12 cm long and the other is 16 cm long The distance between their points of intersection is 4 cm Find the length of the segment connecting the midpoints of the two chords 4 A circle is inscribed in a square with a side length of 8 cm Find the area of the circle 5 A tangent line is drawn to a circle from an external point The distance from the point of tangency to the external point is 12 cm If the radius of the circle is 5 cm find the length of the tangent line Part 4 Proof 10 points Prove that the measure of an inscribed angle is half the measure of its intercepted arc Bonus 5 points A regular hexagon is inscribed in a circle Find the ratio of the area of the hexagon to the area of the circle Answer Key Part 1 Multiple Choice 1 c A circle is a threedimensional shape 2 b Diameter 3 b 60 degrees 4 4 d All of the above 5 c The inscribed angle is half the measure of its intercepted arc 6 b One point 7 d Both b and c 8 b 25 square units 9 c Chord 10 a Two points Part 2 Short Answer 1 a Radius A line segment connecting the center of the circle to a point on the circle b Diameter A line segment passing through the center of the circle and connecting two points on the circle c Chord A line segment connecting two points on the circle d Tangent A line that intersects the circle at exactly one point 2 An inscribed angle is an angle whose vertex lies on the circle and whose sides are chords of the circle A central angle is an angle whose vertex lies at the center of the circle and whose sides are radii of the circle 3 The formula for the area of a circle is A r where A is the area and r is the radius The variable represents the constant ratio of the circumference of a circle to its diameter approximately equal to 314159 4 Two inscribed angles that intercept the same arc are congruent 5 The measure of a central angle that intercepts a semicircle is 180 degrees This is because a semicircle is half of a circle and a central angle that intercepts a whole circle measures 360 degrees Part 3 Problem Solving 1 Circumference C 2r 210 cm 20 cm Area A r 10 cm 100 cm 2 The measure of the intercepted arc is the same as the measure of the central angle which is 120 degrees 3 Let M be the midpoint of the first chord and N be the midpoint of the second chord Since M and N are midpoints MN is parallel to the line segment connecting the endpoints of the two chords Therefore triangle MNO is similar to the larger triangle formed by the two chords and their intersection point Using the properties of similar triangles we can find that MN is 4 5 cm long 4 The diameter of the circle is equal to the side length of the square which is 8 cm Therefore the radius of the circle is 4 cm The area of the circle is A r 4 cm 16 cm 5 Let O be the center of the circle P be the external point and T be the point of tangency Triangle OPT is a right triangle where OT is the radius and PT is the tangent line Using the Pythagorean theorem we can find that PT OP OT 12 5 119 cm Part 4 Proof Let O be the center of the circle A and B be the endpoints of the intercepted arc and C be the vertex of the inscribed angle Case 1 The center of the circle lies on a side of the inscribed angle In this case angle AOC is a central angle that intercepts the same arc as angle ACB Since angle AOC is a central angle its measure is equal to the measure of its intercepted arc Angle ACB is half of angle AOC because the sum of the angles in triangle AOC is 180 degrees and angle AOC is twice the measure of angle ACB Therefore angle ACB is half the measure of its intercepted arc Case 2 The center of the circle lies inside the inscribed angle Draw a diameter through the center of the circle and point C Let D be the other endpoint of the diameter Now we have two inscribed angles ACB and ACD both intercepting the same arc AB Using Case 1 we know that angle ACD is half the measure of arc AB Similarly angle BCD is half the measure of arc AB Therefore angle ACB which is the sum of angles ACD and BCD is also half the measure of arc AB Case 3 The center of the circle lies outside the inscribed angle Draw a diameter through the center of the circle and point C Let D be the other endpoint of the diameter Now we have two inscribed angles ACB and ACD both intercepting the same arc AB Using Case 1 we know that angle ACD is half the measure of arc AB Similarly angle BCD is half the measure of arc AB Therefore angle ACB which is the difference between angles ACD and BCD is also half the measure of arc AB Conclusion In all three cases we have shown that the measure of an inscribed angle is half the measure of its intercepted arc Bonus 6 The area of a regular hexagon is given by 332 s where s is the length of a side Since the hexagon is inscribed in a circle its side length is equal to the radius of the circle Therefore the area of the hexagon is 332 r The area of the circle is r The ratio of the area of the hexagon to the area of the circle is 332 r r 332 0827 Therefore the ratio of the area of the hexagon to the area of the circle is approximately 0827