Name: Class: Date: ID: A Math 9 Chapter 8 Practice Test Short Answer 1. O is the centre of this circle and point Q is a point of tangency. Determine the value of t. If necessary, give your answer to the nearest tenth. 2. O is the centre of this circle and point Q is a point of tangency. Determine the values of d and e. If necessary, give your answers to the nearest tenth. 1
Name: ID: A 3. O is the centre of this circle. Determine the value of m. 4. O is the centre of this circle. Determine the value of z. 5. A circle has radius 8 cm. Which of the following measures could NOT be the length of a chord in the circle: 1 cm, 13 cm, 16 cm, or 19 cm? 6. O is the centre of the circle. Determine the value of x to the nearest tenth, if necessary. 2
Name: ID: A 7. O is the centre of this circle. Which line is a tangent? 8. O is the centre of the circle. Determine the value of s to the nearest tenth, if necessary. 9. Point O is the centre of this circle. Determine the values of c and d. 3
Name: ID: A 10. O is the centre of this circle. Determine the value of g. 11. O is the centre of the circle. Determine the value of z to the nearest tenth, if necessary. 12. Point O is the centre of this circle. Without solving for a, sketch and label the length of any extra line segments you need to draw to determine the value of a. 4
Name: ID: A Problem 13. AC, AE, and CE are tangents to this circle. The points of tangency are: B, F, and D The circle has radius 16. The distance from the centre of the circle to each vertex of the triangle is: OC = 42, OA = OE = 29 Determine the side lengths of ACE, to the nearest tenth. 14. A circle has diameter 38 cm. How far from the centre of the circle, to the nearest centimetre, is a chord 26 cm long? 15. Determine the measure of each interior angle of quadrilateral ABCD. 16. AQ is a tangent to the circle with centre B and to the circle with centre C. The points of tangency are P and Q. Determine the value of y to the nearest tenth. 5
Name: ID: A 17. Point O is the centre of the circle. Determine the values of x, y, and z. 6
Math 9 Chapter 8 Practice Test Answer Section SHORT ANSWER 1. ANS: 20.9 PTS: 1 DIF: Moderate REF: 8.1 Properties of Tangents to a Circle 2. ANS: d = 31.2, e = 29 PTS: 1 DIF: Moderate REF: 8.1 Properties of Tangents to a Circle 3. ANS: 50 PTS: 1 DIF: Easy REF: 8.3 Properties of Angles in a Circle 4. ANS: 72 PTS: 1 DIF: Moderate REF: 8.3 Properties of Angles in a Circle 5. ANS: 19 cm PTS: 1 DIF: Easy REF: 8.2 Properties of Chords in a Circle 6. ANS: 8.9 PTS: 1 DIF: Moderate REF: 8.2 Properties of Chords in a Circle 7. ANS: PR PTS: 1 DIF: Easy REF: 8.1 Properties of Tangents to a Circle 8. ANS: 4.9 PTS: 1 DIF: Moderate REF: 8.2 Properties of Chords in a Circle 1
9. ANS: c = 33, d = 114 PTS: 1 DIF: Easy REF: 8.2 Properties of Chords in a Circle 10. ANS: 122 PTS: 1 DIF: Moderate REF: 8.3 Properties of Angles in a Circle 11. ANS: 4.5 PTS: 1 DIF: Easy REF: 8.2 Properties of Chords in a Circle 12. ANS: Answers may vary. For example: PTS: 1 DIF: Easy REF: 8.2 Properties of Chords in a Circle 2
PROBLEM 13. ANS: AC = AB + BC Use the Pythagorean Theorem in OAB and OBC: AB 2 = OA 2 OB 2 and BC 2 = OC 2 OB 2 AB 2 = 29 2 16 2 AB = 29 2 16 2 AB =Ö 24.1867 So, AC =Ö 24.1867 + 38.8329 =Ö 63.0196 BC 2 = 42 2 16 2 BC = 42 2 16 2 BC =Ö 38.8329 AE = AF + FE Use the Pythagorean Theorem in OAF and OEF: AF 2 = OA 2 OF 2 and FE 2 = OE 2 OF 2 AF 2 = 29 2 16 2 AF = 29 2 16 2 AF =Ö 24.1867 So, AE =Ö 24.1867 + 24.1867 =Ö 48.3734 FE 2 = 29 2 16 2 FE = 29 2 16 2 FE =Ö 24.1867 CE = CD + DE Use the Pythagorean Theorem in OCD and ODE: CD 2 = OC 2 OD 2 and DE 2 = OE 2 OD 2 CD 2 = 42 2 16 2 CD = 42 2 16 2 CD =Ö 38.8329 So, CE =Ö 38.8329 + 24.1867 =Ö 63.0196 DE 2 = 29 2 16 2 DE = 29 2 16 2 DE =Ö 24.1867 The triangle has side lengths of about 63, 63, and 48.4. PTS: 1 DIF: Moderate REF: 8.1 Properties of Tangents to a Circle LOC: 9.SS1 TOP: Shape and Space (Measurement) KEY: Problem-Solving Skills 3
14. ANS: Sketch a diagram. Let d represent the distance from the chord to the centre of the circle. Draw a radius from the centre to one end of the chord. Label the known lengths. PR is a chord of the circle, and OQ is perpendicular to the chord, passing through the centre of the circle, so PQ = QR and QR is 1 2 QR = 1 (26 cm) 2 = 13 cm of PR: ST is a diameter of the circle, and OR is a radius of the circle, so OR is 1 2 of ST: ST = 1 (38 cm) 2 = 19 cm Use the Pythagorean Theorem in d 2 + 13 2 = 19 2 OQR. d 2 = 19 2 13 2 d 2 = 192 d = 192 d =Ö 13.8564 So, the chord is approximately 14 cm from the centre of the circle. PTS: 1 DIF: Moderate REF: 8.2 Properties of Chords in a Circle 4
15. ANS: AC is a diameter of the circle, so ABC = 90 and ADC = 90. The sum of the interior angles of a triangle is 180. So, in 46 + 90 + ACB = 180 136 + ACB = 180 ACB = 180 136 ACB = 44 So, BCD = 44 + 27 = 71 The sum of the interior angles of a triangle is 180. So, in 27 + 90 + CAD = 180 117 + CAD = 180 CAD = 180 117 CAD = 63 So, BAD = 63 + 46 = 109 ABC: ACD: So, the interior angles of quadrilateral ABCD have these measures: ABC = 90, BCD = 71, ADC = 90, BAD = 109 PTS: 1 DIF: Moderate REF: 8.3 Properties of Angles in a Circle LOC: 9.SS1 TOP: Shape and Space (Measurement) KEY: Problem-Solving Skills 5
16. ANS: Use the Pythagorean Theorem in AP 2 = 18 2 6 2 ABP to solve for AP. AP = 18 2 6 2 AP =Ö 16.9706 ABP ACQ Consider ACQ as an enlargement of ABP. The scale ratio is: CQ BP = 12 6 = 2 So, AQ = 2(AP) Then, y = AQ AP = 2(AP) AP = AP So, y =Ö 17.0 PTS: 1 DIF: Difficult REF: 8.1 Properties of Tangents to a Circle LOC: 9.SS1 TOP: Shape and Space (Measurement) KEY: Problem-Solving Skills 6
17. ANS: The sum of the central angles in a circle is 360. 130 + 114 + x = 360 244 + x = 360 x = 360 244 x = 116 ACB is an inscribed angle and AOB is a central angle subtended by the same arc. So, ACB = 1 2 AOB y = 1 2 116 y = 58 OA and OB are radii, so AOB is isosceles with OAB = OBA = z. The sum of the angles in a triangle is 180, so in z + z + 116 = 180 2z + 116 = 180 2z = 180 116 2z = 64 z = 64 2 z = 32 AOB: PTS: 1 DIF: Difficult REF: 8.3 Properties of Angles in a Circle LOC: 9.SS1 TOP: Shape and Space (Measurement) KEY: Problem-Solving Skills 7