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MidpointandandDistanceMidpointDistance1-61-6 in the Coordinate Planein the Coordinate PlaneWarm UpLesson PresentationLesson QuizHoltHoltMcDougalGeometryGeometry

1-6Midpoint and Distancein the Coordinate PlaneWarm Up1. Graph A (–2, 3) and B (1, 0).2. Find CD. 83. Find the coordinate of the midpoint of CD.4. Simplify.4Holt McDougal Geometry–2

1-6Midpoint and Distancein the Coordinate PlaneObjectivesDevelop and apply the formula for midpoint.Use the Distance Formula and thePythagorean Theorem to find the distancebetween two points.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneVocabularycoordinate planeleghypotenuseHolt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneA coordinate plane is a plane that isdivided into four regions by a horizontalline (x-axis) and a vertical line (y-axis) .The location, or coordinates, of a point aregiven by an ordered pair (x, y).Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneYou can find the midpoint of a segment byusing the coordinates of its endpoints.Calculate the average of the x-coordinatesand the average of the y-coordinates of theendpoints.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneHolt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneHelpful HintTo make it easier to picture the problem, plotthe segment’s endpoints on a coordinateplane.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 1: Finding the Coordinates of a MidpointFind the coordinates of the midpoint of PQwith endpoints P(–8, 3) and Q(–2, 7). (–5, 5)Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 1Find the coordinates of the midpoint of EFwith endpoints E(–2, 3) and F(5, –3).Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 2: Finding the Coordinates of an EndpointM is the midpoint of XY. X has coordinates(2, 7) and M has coordinates (6, 1). Findthe coordinates of Y.Step 1 Let the coordinates of Y equal (x, y).Step 2 Use the Midpoint Formula:Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 2 ContinuedStep 3 Find the x-coordinate.Set the coordinates equal.Multiply both sides by 2.12 2 x– 2 –2Simplify.Subtract.10 xSimplify.The coordinates of Y are (10, –5).Holt McDougal Geometry2 7 y– 7 –7–5 y

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 2S is the midpoint of RT. R has coordinates(–6, –1), and S has coordinates (–1, 1). Findthe coordinates of T.Step 1 Let the coordinates of T equal (x, y).Step 2 Use the Midpoint Formula:Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 2 ContinuedStep 3 Find the x-coordinate.Set the coordinates equal.Multiply both sides by 2.–2 –6 x 6 64 xSimplify.Add.2 –1 y 1 1Simplify.3 yThe coordinates of T are (4, 3).Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneThe Ruler Postulate can be used to find the distancebetween two points on a number line. The DistanceFormula is used to calculate the distance betweentwo points in a coordinate plane.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 3: Using the Distance FormulaFind FG and JK.Then determine whether FG JK.Step 1 Find thecoordinates of each point.F(1, 2), G(5, 5), J(–4, 0),K(–1, –3)Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 3 ContinuedStep 2 Use the Distance Formula.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 3Find EF and GH. Then determine if EF GH.Step 1 Find the coordinates ofeach point.E(–2, 1), F(–5, 5), G(–1, –2),H(3, 1)Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 3 ContinuedStep 2 Use the Distance Formula.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneYou can also use the Pythagorean Theorem tofind the distance between two points in acoordinate plane. You will learn more about thePythagorean Theorem in Chapter 5.In a right triangle, the two sides that form theright angle are the legs. The side across from theright angle that stretches from one leg to theother is the hypotenuse. In the diagram, a and bare the lengths of the shorter sides, or legs, of theright triangle. The longest side is called thehypotenuse and has length c.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneHolt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 4: Finding Distances in the Coordinate PlaneUse the Distance Formula and thePythagorean Theorem to find the distance, tothe nearest tenth, from D(3, 4) to E(–2, –5).Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 4 ContinuedMethod 1Use the Distance Formula. Substitute thevalues for the coordinates of D and E into theDistance Formula.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 4 ContinuedMethod 2Use the Pythagorean Theorem. Count the units forsides a and b.a 5 and b 9.c2 a2 b2 52 92 25 81 106c 10.3Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 4aUse the Distance Formula and thePythagorean Theorem to find the distance,to the nearest tenth, from R to S.R(3, 2) and S(–3, –1)Method 1Use the Distance Formula. Substitute thevalues for the coordinates of R and S into theDistance Formula.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 4a ContinuedUse the Distance Formula and thePythagorean Theorem to find the distance,to the nearest tenth, from R to S.R(3, 2) and S(–3, –1)Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 4a ContinuedMethod 2Use the Pythagorean Theorem. Count the units forsides a and b.a 6 and b 3.c2 a2 b2 62 32 36 9 45Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 4bUse the Distance Formula and thePythagorean Theorem to find the distance,to the nearest tenth, from R to S.R(–4, 5) and S(2, –1)Method 1Use the Distance Formula. Substitute thevalues for the coordinates of R and S into theDistance Formula.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 4b ContinuedUse the Distance Formula and thePythagorean Theorem to find the distance,to the nearest tenth, from R to S.R(–4, 5) and S(2, –1)Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 4b ContinuedMethod 2Use the Pythagorean Theorem. Count the units forsides a and b.a 6 and b 6.c2 a2 b2 62 62 36 36 72Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 5: Sports ApplicationA player throws the ballfrom first base to a pointlocated between thirdbase and home plate and10 feet from third base.What is the distance ofthe throw, to the nearesttenth?Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneExample 5 ContinuedSet up the field on a coordinate plane so that homeplate H is at the origin, first base F has coordinates(90, 0), second base S has coordinates (90, 90), andthird base T has coordinates (0, 90).The target point P of the throw has coordinates (0, 80).The distance of the throw is FP.Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneCheck It Out! Example 5The center of the pitchingmound has coordinates(42.8, 42.8). When apitcher throws the ball fromthe center of the mound tohome plate, what is thedistance of the throw, tothe nearest tenth? 60.5 ftHolt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneLesson Quiz: Part I1. Find the coordinates of the midpoint of MN withendpoints M(-2, 6) and N(8, 0). (3, 3)2. K is the midpoint of HL. H has coordinates (1, –7),and K has coordinates (9, 3). Find the coordinatesof L. (17, 13)3. Find the distance, to the nearest tenth, betweenS(6, 5) and T(–3, –4). 12.74. The coordinates of the vertices of ABC are A(2, 5),B(6, –1), and C(–4, –2). Find the perimeter of ABC, to the nearest tenth. 26.5Holt McDougal Geometry

1-6Midpoint and Distancein the Coordinate PlaneLesson Quiz: Part II5. Find the lengths of AB and CD and determinewhether they are congruent.Holt McDougal Geometry

Holt McDougal Geometry 1-6 Midpoint and Distance in the Coordinate Plane Set up the field on a coordinate plane so that home plate H is at the origin, first base F has coordinates (90, 0), second base S has coordinates (90, 90), and third base T has coordinates (0, 90). The target point P of the throw has coordinates (0, 80).