Midpoint and Distance Formulas
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1 Midpoint and Distance Formulas Find the midpoint of a segment on the coordinate plane. Find the distance between two points on the coordinate plane. Fremont are the Midpoint and Distance Formulas used in emergenc medicine? MAHA 9 Wahoo Mis s our i Ri e v A square grid is superimposed on a map of eastern Nebraska where emergenc medical assistance b helicopter is available from both Lincoln and maha. Each side of a square represents 0 miles. You can use the formulas in this lesson to determine whether the site of an emergenc is closer to Lincoln or to maha. r Nebraska LINCLN THE MIDPINT FRMULA Recall that point M is the midpoint of segment PQ if M is between P and Q and PM MQ. There is a formula for the coordinates of the midpoint of a segment in terms of the coordinates of the endpoints. You will show that this formula is correct in Eercise 4. Midpoint Formula Stud Tip Words Midpoints The coordinates of the midpoint are the means of the coordinates of the endpoints. If a line segment has endpoints at (, and (,, then the midpoint of the segment has Model (, coordinates 冢,. Smbols ( midpoint 冢, (, Eample Find a Midpoint LANDSCAPING A landscape design includes two square flower beds and a sprinkler halfwa between them. Find the coordinates of the sprinkler if the origin is at the lower left corner of the grid. The centers of the flower beds are at (4, 5 and (4, 3. The sprinkler will be at the midpoint of the segment joining these points ,, 8 8, or (9, 9 冢 冢 冢 The sprinkler will have coordinates (9, 9. 4 Chapter 8 Conic Sections Lawn Flowers Ground cover Flowers Paved Patio
2 THE DISTANCE FRMULA Recall that the distance between two points on a number line whose coordinates are a and b is a b or b a. You can use this fact and the Pthagorean Theorem to derive a formula for the distance between two points on a coordinate plane. Suppose ( and (, name two points. Draw a right triangle with vertices at these points and the point (,. The lengths of the legs of the right triangle are and. Let d represent the distance between ( and (,. Now use the Pthagorean Theorem. ( (, (, d c a b Pthagorean Theorem d Replace c with d, a with, and b with. d ( ( ( ; ( d ( ( Find the nonnegative square root of each side. Stud Tip Distance In mathematics, distances are alwas positive. Words The distance between two points with coordinates ( and (, is given b Model Distance Formula (, d ( (. ( d ( ( Eample Find the Distance Between Two Points What is the distance between A( 3, 6 and B(4, 4? d ( ( Distance Formula [4 3] ( ( 4 6 Let ( ( 3, 6 and (, (4, ( or 49 Subtract. Simplif. The distance between the points is 49 units.
3 A Follow-Up of Lesson 8- Midpoint and Distance Formulas in Three Dimensions You can derive a formula for distance in three-dimensional space. It ma seem that the formula would involve a cube root, but it actuall involves a square root, similar to the formula in two dimensions. Suppose (, z and (,, z name two points z in space. Draw the rectangular bo that has opposite (, z vertices at these points. The dimensions of the bo are,, and z z. Let a be the d length of a diagonal of the bottom of the bo. B the Pthagorean Theorem, a. a To find the distance d between (, z and (,, z, appl the Pthagorean Theorem to the ( right triangle whose legs are a diagonal of the,, z bottom of the bo and a vertical edge of the bo. d a z z Pthagorean Theorem d z z a d ( ( (z z d ( ( (z z (, and so on Take the square root of each side. The distance d between the points with coordinates (, z and (,, z is given b the formula d ( ( (z. z Eample Find the distance between (, 0, 3 and (4,, 9. d ( ( (z z Distance Formula (4 ( 0 [9 ( 3] 5 or 38 The distance is 38 or about.33 units. (, z (, 0, 3 (,, z (4,, 9 (4,, (, 0, 3 z In three dimensions, the midpoint of the segment with coordinates (, z and z (,, z has coordinates,, z. Notice how similar this is to the Midpoint Formula in two dimensions. (continued on the net page Investigating Slope-Intercept Form 47 Algebra Activit Midpoint and Distance Formulas in Three Dimensions 47
4 Conic Sections Write equations of conic sections in standard form. Identif conic sections from their equations. can ou use a flashlight to make conic sections? Recall that parabolas, circles, ellipses, and hperbolas are called conic sections because the are the cross sections formed when a double cone is sliced b a plane. You can use a flashlight and a flat surface to make patterns in the shapes of conic sections. parabola circle ellipse hperbola Stud Tip Reading Math In this lesson, the word ellipse means an ellipse that is not a circle. STANDARD FRM The equation of an conic section can be written in the form of the general quadratic equation A B C D E F 0, where A, B, and C are not all zero. If ou are given an equation in this general form, ou can complete the square to write the equation in one of the standard forms ou have learned. Conic Section Parabola Standard Form of Conic Sections Standard Form of Equation a( h k or a( k h Circle ( h ( k r Ellipse ( a h ( b k or ( a k ( b h, a b Hperbola ( a h ( b k or ( a k ( b h Eample Rewrite an Equation of a Conic Section Write the equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation. Write the equation in standard form riginal equation Isolate terms Complete the square. ( ( 3 ( Divide each side b 6. The graph of the equation is an ellipse with its center at (3, 0. ( Lesson 8-6 Conic Sections 449
5 IDENTIFY CNIC SECTINS Instead of writing the equation in standard form, ou can determine what tpe of conic section an equation of the form A B C D E F 0, where B 0, represents b looking at A and C. Conic Section Parabola Circle Identifing Conic Sections Relationship of A and C A 0 or C 0, but not both. A C Ellipse A and C have the same sign and A C. Hperbola A and C have opposite signs. Eample Analze an Equation of a Conic Section Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola. a A and C. Since A and C have opposite signs, the graph is a hperbola. b A 4 and C 4. Since A C, the graph is a circle. c C. Since there is no term, A 0. The graph is a parabola.
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