2.1 Length of a Line Segment
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1 .1 Length of a Line Segment MATHPOWER TM 10 Ontario Edition pp To find the length of a line segment joining ( 1 y 1 ) and ( y ) use the formula l= ( ) + ( y y ). 1 1 Name An equation of the circle with centre O(0 0) and radius r is + y = r. 1. Determine the length of the line segment joining each pair of points. Epress each length as an eact solution and as an approimate solution to the nearest tenth. a) ( 7) and ( 1 ). Communication Eplain why POR is a right triangle. y P( ) O(0 0) 0 b) (0 ) and (6 10) R(4 4). Determine the radius of the circle with centre ( 6) and point ( 7) on its circumference. Round the radius to the nearest tenth if necessary.. Classify each triangle as equilateral isosceles or scalene. Then find each perimeter to the nearest tenth. a) W( ) X( 1 ) Y( ) ( ) b) A B( 4 1) C( 1) 6. The vertices of a right triangle are ( ) ( 8) and ( 4). Find the area of the triangle. 7. Three points A( 1) B( 4) and C(7 7) lie on a straight line. Show that B is the midpoint of AC. 8. The coordinates of the endpoints of the diameter of a circle are ( ) and ( ). Find the length of the radius of the circle. 4. Find the perimeter of parallelogram ABCD. y A( ) B(6 ) 9. a) Verify that the quadrilateral with vertices D( 6) E( ) F( 1) and G( 4 4) is a rectangle. b) Determine the length of its diagonals to the nearest tenth. 0 D( ) C( ) Copyright 001 McGraw-Hill Ryerson Limited Chapter 1
2 . Midpoint of a Line Segment Name MATHPOWER TM 10 Ontario Edition pp To find the midpoint M of a line segment joining ( 1 y 1 ) and ( y ) use the midpoint formula 1 + y1 + y. 1. Determine the midpoint of each line segment with the given endpoints. a) ( 6 ) and (4 8) b) (1. ) and ( 6.) 4. The endpoints of the diameter of a circle are ( 11) and ( 9). What are the coordinates of the centre of the circle? c) ( ) and (0 600) d) 7 1 and 4 e) (a b) and ( a b) 4. The endpoints of line segment MN are M( 6 10) and N( ). Find the coordinates of the point P on the line segment MN such that MP:PN = :1. f) ( 6a b) and (11a 0). Find the midpoints of the sides of DEF. D( 4) y 6. A square has vertices K( 4 ) L( 4) M(4 ) and N( 4). a) Find the coordinates of the midpoint of each side. F( 8 4) 0 E( 7) b) Find the coordinates of the point of intersection of the diagonals.. Communication One endpoint of a line segment is D( 7). The midpoint of the line segment is M(. 1.). Eplain how to find the coordinates of the other endpoint E of the line segment. c) Find the perimeter of the square formed by joining the midpoints of the sides of square KLMN. 7. Verte V of UVW has coordinates (4 6). The coordinates of the midpoint of UV are (1 6) and the coordinates of the midpoint of VW are ( ). Find the coordinates of points U and W. Copyright 001 McGraw-Hill Ryerson Limited Chapter
3 Name.4 Verifying Properties of Geometric Figures MATHPOWER TM 10 Ontario Edition pp The following formulas can be used to determine characteristics of geometric figures and to verify geometric properties. Slope of a line segment: m y y = 1 Length of a line segment: l= ( 1) + ( y y1) Midpoint of a line segment: 1 + y1 + y Point-slope form of the equation of a line: y y 1 = m( 1 ) Slope and y-intercept form of the equation of a line: y = m + b 1. Communication For any three points A B and C not in a line M and N are the midpoints of AB and AC respectively. How can you prove that MN BC and MN = 1 BC? 4. ABC has vertices A( ) B( ) and C( ). a) Find the equations of the three altitudes. b) Find the intersection point of any two of the altitudes. c) Verify that this point (the orthocentre of the triangle) is on the altitude not used in part b).. DEF has vertices D( 1 ) E(7 1) and F(4 6). Classify the triangle as a) isosceles or scalene b) right-angled or not. The sides of a triangle have the equations y + 1 = 0 + y = 0 and + y 6 = 0. Verify that the triangle is an isosceles right triangle.. The vertices of a quadrilateral are S(1 ) T( ) U(6 7) and V(4 4). Verify each of the following. a) STUV is a parallelogram. b) The diagonals of STUV bisect each other. c) STUV is a rhombus. d) The diagonals of STUV are perpendicular to each other. 6. A quadrilateral has vertices P( 1) Q( 7) R(9 ) and S( 1 1). a) Verify that PQRS has no equal sides and no parallel sides. b) Find the midpoints A B C and D of PQ QR RS and SP. c) Verify that ABCD is a parallelogram. 4 Chapter Copyright 001 McGraw-Hill Ryerson Limited
4 . Distance From a Point to a Line Name MATHPOWER TM 10 Ontario Edition pp To determine the distance from a given point to a line whose equation is given a) write an equation for the perpendicular from the given point to the given line b) find the coordinates of the point of intersection of the perpendicular and the given line c) use the distance formula 1. Communication Eplain how to find the shortest distance from the point P( 7) to the line =. 4. Find the shortest distance from the given point to the given line. Round to the nearest tenth if necessary. a) (0 0) and y = +. In each case write an equation for the line that is perpendicular to the line with the given equation and passes through the given point. a) 1 y = 7;( ) b) (0 0) and 1 8y 9 = 0 c) ( 4 ) and y = 4 b) y = ; ( 1 7) d) (6 ) and 7 + y + = 0 c) 4 y 7 = 0; ( ) d) + y + = 0; ( 4). A line has a y-intercept of and an -intercept of 4. What is the shortest distance from the origin to this line?. Find the eact value of the shortest distance from the given point to the given line. a) (0 0) and y = a) Find the eact distance from the point A( 7) to the line joining B( 1) and C(4 ). b) Find the eact length of BC. b) (0 0) and + 1y 9 = 0 c) ( ) and 1 y = + 9 c) Use your answers to parts a) and b) to find the area of ABC. d) ( ) and 4 + y + 14 = 0 Copyright 001 McGraw-Hill Ryerson Limited Chapter
5 Answers CHAPTER Analytic Geometry.1 Length of a Line Segment 1. a) 160; 1. 6 b) a) isosceles; 17.7 b) equilateral; length of PQ = 18 length of QR = length of PR = 0; PQ + QR = 18 + = 0 = PR Thus by the Pythagorean Theorem PQR is a right triangle square units 61; AB = 4 ; BC = a) DE = GF = 10 FE = GD = 40 DE + DG = = 0 = GE Thus EDG = 90. b) 7.1. Investigation: Midpoints of Horizontal and Vertical Line Segments 1. a) (1 4) b) ( 4. 7) c) ( 0..) d) (1 4). a) ( 7 ) b) (. 0) c) (6.4 ) d) ( 4 ) ( 7 ) and ( 10 ) 4. Answers may vary. (8 ) and ( 4 ) or (0 ) and (4 ). There are an infinite number of pairs of coordinates on the line y = that are on opposite sides of M( ).. a) ( ) b) ( 9 1.) c) (0. 0.) d) ( 14 1) 6. a) (1 6) b) (0 1) c) ( 7.) d) (1 1) (1 ) and (1 7) 8. Answers may vary. ( 0) and ( 10) or ( ) and ( 1). There are an infinite number of pairs of coordinates on the line = that are on opposite sides of M( ).. Midpoint of a Line Segment 1. a) ( 1 ) b) (. 0.) c) (7 0) d) 1 1 e) (0.b) f) a b. ( 0) (1. 1.) ( 1..). Let the coordinates of E be ( 1 y 1). Substitute the values into the midpoint formula 1 + y1 + ( 7) (. 1. ) =. Then solve for 1 and y 1 using these equations: 1 + =. and y1 + ( 7) = 1..( 1 y1)= ( 10) 4. ( 0. 10). P(0 4) 6. a) ( 0..) (. 0.) (0..) (. 0.) b) (0 0) c) 0 7. U( 6) W( ).4 Verifying Properties of Geometric Figures 1. Find the coordinates of M and N using the midpoint formula. Find the slope of MN and of BC. If the slopes are equal the line segments are parallel. Find the length of MN and of BC using the length formula. The length of MN should be half the length of BC.. a) isosceles b) right. a) slopes of ST and VU = slopes of SV and ; TU = b) The midpoint of both SU and TV is c) ST = TU = UV = VS = 1 d) slope of TV = 1; slope of SU = 1 4. a) from A to BC: y = 4; from B to AC: 1 9 y = + from C to AB: y = + ; b) c) satisfies the equation of the other altitude 7 7 Copyright 001 McGraw-Hill Ryerson Limited Chapter 7
6 . vertices: A( ) B(1 ) C( 4 6); AB = AC = 1; slope of AB = slope of AC = 6. a) PQ = 7 slope = 1; QR = slope = ; RS = 116 slope = ; SP = 8 slope = 1 b) A(0 4) B(6 ) C(4 1) D( 0) c) slopes of AB and DC = 1 slopes of AD and 6 ; BC =. Distance From a Point to a Line 1. The shortest distance is along the line through P that is perpendicular to the vertical line =. The length of the line segment joining P( 7) and Q( 7) is 8.. a) y = + 9 or + y 9 = 0 19 b) y = or y 19 = 0 7 c) y = or + 4y + 7 = d) y = or y = 0. a) 0 b) c) 90 d) 4. a) 4. b) 1.7 c).9 d) a) b) c) Chapter Copyright 001 McGraw-Hill Ryerson Limited
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