2 9.6 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55Review §Any QUESTIONS About§9.6 → Exponential Decay & GrowthAny QUESTIONS About HomeWork§9.6 → HW-48
3 The Distance FormulaThe distance between the points (x1, y1) and (x2, y1) on a horizontal line is |x2 – x1|.Similarly, the distance between the points (x2, y1) and (x2, y2) on a vertical line is |y2 – y1|.
4 Pythagorean DistanceNow consider any two points (x1, y1) and (x2, y2).These points, along with (x2, y1), describe a right triangle. The lengths of the legs are |x2 – x1| and |y2 – y1|.
5 Pythagorean DistanceFind d, the length of the hypotenuse, by using the Pythagorean theorem:d2 = |x2 – x1|2 + |y2 – y1|2Since the square of a number is the same as the square of its opposite, we can replace the absolute-value signs with parentheses:d2 = (x2 – x1)2 + (y2 – y1)2
6 Distance Formula Formally The distance d between any two points (x1, y1) and (x2, y2) is given by
7 Example Find Distance Find the distance between (3, 1) and (5, −6). Find an exact answer and an approximation to three decimal places.Solution: Substitute into the distance formulaSubstitutingThis is exact.Approximation
8 Example Verify Rt TriAngle Let A(4, 3), B(1, 4) and C(−2, −4) be three points in the plane. Connect these Dots to form a Triangle, Then:Sketch the triangle ABCFind the length of each side of the triangleShow that ABC is a right triangle.
9 Example Verify Rt TriAngle Soln a. Sketch TriAngle
10 Example Verify Rt TriAngle Soln b. Find the length of each side of the triangle → Use Distance Formula
11 Example Verify Rt TriAngle Soln c.: Show that ABC is a Rt triangle.Check that a2 + b2 = c2 holds in this triangle, where a, b, and c denote the lengths of its sides. The longest side, AC, has length 10 units.It follows from the converse of the Pythagorean Theorem that the triangle ABC IS a right triangle.
12 Example BaseBall Distance The baseball “diamond” is in fact a square with a distance of 90 feet between each of the consecutive bases. Use an appropriate coordinate system to calculate the distance the ball will travel when the third baseman throws it from third base to first base.
13 Example BaseBall Distance Solution: conveniently choose home plate as the origin and place the x-axis along the line from home plate to first base and the y-axis along the line from home plate to third base
14 Example BaseBall Distance Find from the DiagramThe coordinates of home plate (O), first base (A) second base (C) and third base (B)
15 Example BaseBall Distance Find the distance between points A & B127.3 ft
16 The MidPoint FormulaNow that we have derived the Distance formula from the Pythagorean Theorem we use the distance formula to develop a formula for the coordinates of the MidPoint of a segment connecting two points.
17 The MidPoint FormulaIf the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint arey(x2, y2)(x1, y1)xThat is, to locate the midpoint, average the x-coordinates and average the y-coordinates
18 Example MidPoint Formula Find the midpoint of the line segment joining the points P(−3, 6) and Q(1, 4)Solution: (x1, y1) = (−3, 6) & (x2, y2) = (1, 4)
19 CIRCLE DefinedA circle is a set of points in a Cartesian coordinate plane that are at a fixed distance r from a specified point (h, k).The fixed distance r is called the radius of the circle, andThe specified point (h, k) is called the center of the circle.
20 CIRCLE GraphedThe graph of a circle with center (h, k) and radius r.
21 CIRCLE - EquationThe equation of a circle with center (h, k) and radius r isThis equation is also called the standard form of an equation of a circle with radius r and center (h, k).
22 Example Find Circle Eqn Find the center-radius form of the equation of the circle with center (−3, 4) and radius 7.Solution:
23 Example Graph Circle Graph each equation Solution: Center: (0, 0) Radius: 1Called the unit circle
24 Example Graph CircleSolution:Center: (−2, 3)Radius: 5
25 Equation ↔ Circle Note that stating that the equation: represents the circle of radius 5 with center (–3, 4) means two things:If the values of x and y are a pair of numbers that satisfy the equation, then they are the coordinates of a point on the circle with radius 5 and center (–3, 4).If a point is on the circle, then its coordinates satisfy the equation
26 Circle Eqn → General Form The general form of the equation of a circle is
27 Example General FormFind the center and radius of the circle with equation x2 +y2 − 6x + 8y +10 = 0Solution: COMPLETE the SQUARE for both x & yCenter: (3, – 4) Radius:
28 Example General FormFind the center & radius and then graph the circle x2 + y2 + 2x – 6y + 6 = 0Solution: Complete Square for both x & y to convert to Standard Formx2 + 2x + y2 – 6y = –6x2 + 2x y2 – 6y + 9 = –(x + 1)2 + (y – 3)2 = 4(x – (–1))2 + (y – 3)2 = 2 2
29 Example General Form Solution: Graph Sketch Graph ySolution: GraphCenter: (–1, 3)Radius: 2Sketch Graph(–1, 3)x(x – (–1))2 + (y – 3)2 = 2 2
30 WhiteBoard Work Problems From §10.1 Exercise Set Circle Eqns 16, 26, 38, 48, 54, 56Circle Eqns
31 Circle as Conic Section All Done for TodayCircle as Conic Section