Math 9 Unit 8: Circle Geometry Pre-Exam Practice Name: 1. A Ruppell s Griffon Vulture holds the record for the bird with the highest documented flight altitude. It was spotted at a height of about 11 km above the Earth s surface. The radius of Earth is approximately 6400 km. How far was the vulture from the horizon, H? Calculate this distance to the nearest kilometre. H V 11 km 6400 km 2. AC, AE, and CE are tangents to this circle. The points of tangency are: B, F, and D. The circle has radius 11. The distance from the centre of the circle to each vertex of the triangle is: C = 34, A = E = 19 Determine the side lengths of ACE, to the nearest tenth. A B F 19 19 11 11 34 C D E 3. 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. C 12 B 18 6 Q y P A 4. A circle has diameter 32 cm. How far from the centre of the circle, is a chord 20 cm long? 5. A pedestrian underpass is constructed using a cylindrical pipe of radius 2.6 m. The bottom of the pipe will be filled and paved. The headroom at the centre of the path is 3.9 m. How wide is the path to the nearest tenth of a metre? 2.6 m 3.9 m
6. Determine the values of x and y. What can you say about line segment AD? A B 53 y 37 C x D 7. Point is the centre of the circle. Determine the values of x, y, and z. A z 133 x 107 B y C 8. Circle C has two chords drawn in it. Chord XY is 9.3 cm long and chord YZ is 12.4 cm long. When X and Z are joined by a straight line, the line runs through the centre of the circle. What is the diameter of the circle? Sketch and label the scenario. Justify your calculations. 9. Circle C has two inscribed angles measuring 42 and 36. What is the measure of STU, TUV and SRT? 10. The diagram shown below is a semicircle with inscribed angles. Point C is the centre point of the complete circle. What is the measure of DBE, BDE and ADB? 11. The radius of a circle is 90 mm long and passes through the centre of a chord at a distance of 46 mm from the circumference of the circle. What is the length of the chord to the nearest hundredth? Show your thinking.
12. A circular garden has a diameter of 6 m. There is a tree planted in the centre. David wants to build a wall in the garden. He needs to build the wall so that the distance from the edge of the garden to the centre of the wall is of the distance from the centre of the garden to the edge of the garden. Determine the length of the wall, to the nearest tenth, if both ends touch the edge of the garden. 13. Circle C has a diameter of 100 mm. What is the measure of the chord AB? Express your answer to the nearest whole millimetre. 14. Determine the length of chord PR in the figure below. Show your work, to the nearest whole millimetre. 15. In the figure shown, what is the measure of PS? 16. In the figure below, line QP is a tangent to the circle, and has a length of 21 cm. Line Q runs from the centre of the circle to the end of the tangent and is 35 cm long. What is the diameter of the circle?
17. Given the information in the diagram, what is the measure of the diameter of the circle? 18. Ravinder is flying a model plane at the end of a tether. The plane breaks off the tether at point A and lands 24 m away at point B. How far is the plane from Ravinder? Justify your response. 19. A round dining table has a centre, H, and a diameter of 1.2 m. The table is pushed into a corner of a dining room, as shown. Determine the distance between the corner of the wall, C, and the edge of the table. Express your answer to the nearest millimetre. 20. Four round patio stones are placed together as shown. Each patio stone has a diameter of 30 cm. What would be the size of the largest possible square tile that could be placed in the middle of the arrangement of patio stones? Express your answer to the nearest centimetre.
Answer Section 1. ANS: V H Use the Pythagorean Theorem in solve for HV. to 6400 km 6411 km The vulture was about 375 kilometres from the horizon. 2. ANS: AC = AB + BC Use the Pythagorean Theorem in AB and BC: and So, AE = AF + FE Use the Pythagorean Theorem in AF and EF: and So, CE = CD + DE Use the Pythagorean Theorem in CD and DE:
and So, 3. ANS: Use the Pythagorean Theorem in to solve for AP. Consider as an enlargement of. The scale ratio is: So, AQ = 2(AP) Then, So, y 17.0 4. 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. S P 20 cm Q d 32 cm R T Label the known lengths. PR is a chord of the circle, and Q is perpendicular to the chord, passing through the centre of the circle, so PQ = QR and QR is of PR:
ST is a diameter of the circle, and R is a radius of the circle, so R is of ST: Use the Pythagorean Theorem in. Q 10 cm R d 16 cm So, the chord is approximately 12 cm from the centre of the circle. 5. ANS: Draw a radius from the centre of the pipe,, to an edge of the path, E. Label the midpoint of the path F. E is a radius, so: E = 2.6 m Use the Pythagorean Theorem in solve for EF. to E 2.6 m 1.3 m F The width of the path is twice the length of EF. So, the width of the path is about 4.5 m. 6. ANS: Since and are inscribed angles subtended by the same arc AB, they are congruent. x = 37 The sum of the interior angles of a triangle is 180. So, in :
Since arc AD subtends right, arc AD is a semicircle and line segment AD is a diameter of the circle. 7. ANS: The sum of the central angles in a circle is 360. is an inscribed angle and angle subtended by the same arc. So, is a central A z 133 120 107 B y A and B are radii, so is isosceles with. The sum of the angles in a triangle is 180, so in : C 8. ANS: Since line XZ runs through the centre of the circle, XZ is a diameter. XYZ is 90º. Using the Pythagorean theorem, the length of XZ The diameter of the circle is 15.5 cm.
9. ANS: a) STU and SVU are both subtended by arc SU. Therefore, SVU = STU. STU is 36. b) TUV and TSV are both subtended by arc VT. Therefore, TUV = TSV. TUV is 42. c) STU, TSV, and SRT form a triangle. SRT = 180 (42 + 36) = 102 SRT is 102. 10. ANS: a) DBE and DAE are both subtended by arc DE. Therefore, DBE = DAE. DBE is 35. b) BDE and BAE are both subtended by arc BE. Therefore, BDE = BAE. BDE is 30. c) ADB is subtended by diameter AB. ADB is 90. 11. ANS: Chord MN has a length of 157.02 mm. 12. ANS: Diameter of garden = 6 m Radius of garden = 3 m Length of VZ = 0.6 3 = 1.8 m Length UZ = 3.0 1.8 = 1.2 m Triangle UYZ has a hypotenuse of 3 m and one side 1.2 m long.
Length of wall is The wall is approximately 5.5 m long. 13. ANS: The length of chord AB is approximately 71 mm. 14. ANS: NPQ is a right angled triangle. PR = PQ (Both are legs of an equilateral triangle.)
PR = 48 mm Chord PR is 48 mm long. 15. ANS: PQ = 90 PQ = 180 (90 56 = 34 Since PS is an isosceles triangle, SP = PS, so PS is 17 PS is 17 16. ANS: Line P is a radius of the circle. Diameter = 2(radius) = 2 (28) = 56 cm The diameter of circle is 56 cm. 17. ANS: Radius = T Diameter = 2 48 = 96 The diameter of the circle is 96 mm. 18. ANS: The plane lands 30 m away from Ravinder.
19. ANS: The distance from the corner of the dining room to the edge of the table is approximately 249 mm. 20. ANS: he largest possible square tile would measure 12 cm by 12 cm.