§10.1 Distance MIdPoint Eqns Chabot Mathematics §10.1 Distance MIdPoint Eqns Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

9.6 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §9.6 → Exponential Decay & Growth Any QUESTIONS About HomeWork §9.6 → HW-48

The Distance Formula The 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|.

Pythagorean Distance Now 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|.

Pythagorean Distance Find d, the length of the hypotenuse, by using the Pythagorean theorem: d2 = |x2 – x1|2 + |y2 – y1|2 Since 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

Distance Formula Formally The distance d between any two points (x1, y1) and (x2, y2) is given by

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 formula Substituting This is exact. Approximation

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 ABC Find the length of each side of the triangle Show that ABC is a right triangle.

Example  Verify Rt TriAngle Soln a. Sketch TriAngle

Example  Verify Rt TriAngle Soln b. Find the length of each side of the triangle → Use Distance Formula

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.

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.

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

Example  BaseBall Distance Find from the DiagramThe coordinates of home plate (O), first base (A) second base (C) and third base (B)

Example  BaseBall Distance Find the distance between points A & B 127.3 ft

The MidPoint Formula Now 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.

The MidPoint Formula If the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are y (x2, y2) (x1, y1) x That is, to locate the midpoint, average the x-coordinates and average the y-coordinates

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)

CIRCLE Defined A 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, and The specified point (h, k) is called the center of the circle.

CIRCLE Graphed The graph of a circle with center (h, k) and radius r.

CIRCLE - Equation The equation of a circle with center (h, k) and radius r is This equation is also called the standard form of an equation of a circle with radius r and center (h, k).

Example  Find Circle Eqn Find the center-radius form of the equation of the circle with center (−3, 4) and radius 7. Solution:

Example  Graph Circle Graph each equation Solution: Center: (0, 0) Radius: 1 Called the unit circle

Example  Graph Circle Solution: Center: (−2, 3) Radius: 5

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

Circle Eqn → General Form The general form of the equation of a circle is

Example  General Form Find the center and radius of the circle with equation x2 +y2 − 6x + 8y +10 = 0 Solution: COMPLETE the SQUARE for both x & y Center: (3, – 4) Radius:

Example  General Form Find the center & radius and then graph the circle x2 + y2 + 2x – 6y + 6 = 0 Solution: Complete Square for both x & y to convert to Standard Form x2 + 2x + y2 – 6y = –6 x2 + 2x + 1 + y2 – 6y + 9 = –6 + 1 + 9 (x + 1)2 + (y – 3)2 = 4 (x – (–1))2 + (y – 3)2 = 2 2

Example  General Form Solution: Graph Sketch Graph y Solution: Graph Center: (–1, 3) Radius: 2 Sketch Graph (–1, 3) x (x – (–1))2 + (y – 3)2 = 2 2

WhiteBoard Work Problems From §10.1 Exercise Set Circle Eqns 16, 26, 38, 48, 54, 56 Circle Eqns

Circle as Conic Section All Done for Today Circle as Conic Section

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

ReCall Logarithmic Laws Solving Logarithmic Equations Often Requires the Use of the Properties of Logarithms