The Distance Between Any Two Points



The distance between points A(x1, y1) and B(x2, y2)in the coordinate plane is A line segment is the part of a line between two specific points, including the points themselves. The slope of a line segment is the same as the slope of the line containing the line segment.Parallel line segments have the ________________ slope. The slope of two perpendicular line segments are _______________________________.Example 1 Find the length of the line segments with these end points. a) A(-1, 0) and B(5, 2)b) G(-7, 8) and H(-7, -5)Example 2 An airplane at coordinates (150, 136), which is heading for Sudbury (0, 0), has to be diverted from poor weather conditions to either North Bay (85, -10) or Timmins (-10, 155). If the airplane is carrying enough fuel to get to Sudbury, which alternate airport would it be safe to head for?Example 3 A triangle has vertices at A(-1, -1), B(2, 0), and C(1, 3). Find the lengths and slopes of the sides of the triangle. What kind of triangle is it?Definitions:ScaleneIsoscelesEquilateralRight TriangleVerticalHorizontalThe Distance Between Any Two PointsHomeworkFor each pair of points, Draw the line segment joining the pointsCalculate the length of the line segmentCalculate the slope of the line segmentA(2, 6) and B(5, 2) b) C(-3, 4) and D(3, 2)c) E(-6, 8) and F(-6, -9)d) M(1, 10) and N(20, 190)e) K(-10, -2) and L(6, -2)Which of these points is closest to point A(-3.2, 5.6): B(1.8, -4.3), C(0.7, 8.9), or D(-7.6, 3.9)a) Which of the lines in question 1 are vertical? How can you tell from the coordinates of the end points that a line segment is vertical? How can you quickly calculate the length of a vertical line segment if you know the coordinates of its end points?Which of the lines in question 1 are horizontal? How can you tell from the coordinates of the end points that a line segment is horizontal? How can you quickly calculate the length of a horizontal line segment if you know the coordinates of its end points?The following points are the vertices of triangles. Draw each triangle on a coordinate grid. Predict if the triangle is scalene, isosceles, or equilateral. Calculate the length of each side to check your prediction.a) A(3, 3), B(-1, 2), C(0, -2)b) D(2, -3), E(-2, -4), F(6, -6)c) G(-1, 3), H(-2, -2), I(2, 0)Do points P(-2, -3), Q(4, 1), and R(2, 4) form a right triangle? Justify your answer. How can you find the answer by calculation alone?Without drawing diagrams, explain which sets of points are the vertices of right triangles.a) S(-2, 2), T(-1, -2), U(7, 0) b) X(3, 2), Y(1, -2), Z(-3, 6)Extension:How does the formula for the distance between two points in the plane differ from the formula for the distance of a point from the origin? You found the equation of a circle with centre at the origin by setting the distance from the origin equal to the radius of the circle. How could you use the formula for the distance between two points in the plane to find the equation of a circle whose centre is not the origin?Find the coordinates of three points that are 5 units from point C(2, 3). Develop a rule for finding the coordinates of all points that are 5 units from C.Answers:1. a) 5, -43 b) 6.32, -13 c) 4.12, undefined d) 181, 18019 f) 16, 0 2. D3. a) c; x-values are same; subtract y-values b) e; y-values are same, subtract x-values4. a) isosceles b) scalene c) scalene 5. yes; the sum of the square of the hypotenuse is equal to the sum of the squares of the other two sides. 6. a) yes b) no ................
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