<p> Lesson 9.5: Distance in Coordinate Geometry</p><p>In this lesson you will:  learn a formula for finding the distance between two points on a coordinate plane  discover the general equation of a circle</p><p>Viki is standing on the corner of Seventh Street and 8th Avenue, and her brother Scott is on the corner of Second Street and 3rd Avenue. To find her shortest sidewalk route to Scott, Viki can simply count blocks. But if Viki wants to know her diagonal distance to Scott, she would need the ______Theorem to measure across blocks.</p><p>You can think of a coordinate plane as a grid of streets with two sets of parallel lines running perpendicular to each other. Every segment in the plane that is not in the x- or y-direction is the ______of a right triangle whose legs are in the x- and y- directions. So you can use the Pythagorean Theorem to find the distance between any two points on a coordinate plane.</p><p>Investigation 9.5: “The Distance Formula” In this investigation, we will find the length of each segment on a coordinate grid by using the segment as the hypotenuse of a right triangle. A.) On the grids below, use the segment as the hypotenuse of a right triangle. Draw the legs along the grid lines and label their lengths. </p><p>B.) B.) Find the length of each segment using the Pythagorean Theorem. </p><p>Graph a: </p><p>Graph b:</p><p>Graph c:</p><p>Graph d:</p><p>What if the points are so far apart that it’s not practical to plot them? For example, what is the distance between the points A(15,34) and B(42,70)? A formula that uses the coordinates of the given points would y be helpful. To find this formula, you first need to find the lengths of the legs in terms of the x- and y-coordinates. From your work with slope x triangles, you know how to calculate horizontal and vertical distances. </p><p>C.) How would you find the horizontal distance (labeled x) in the right triangle above?</p><p>D.) Write an expression for the length of the horizontal leg of any right triangle with the coordinates (x1, y1) and (x2, y2).</p><p>E.) How would you find the vertical distance (labeled y) in the right triangle above?</p><p>F.) Write an expression for the length of the vertical leg of any right triangle using the coordinates (x1, y1) and (x2, y2).</p><p>G.) Use your work in parts C-F, and the Pythagorean Theorem, to find the distance between points A(15,34) and B(42,70). (AB)2 = </p><p>H.) Generalize what you have learned about the distance between two points in a coordinate plane by completing the conjecture below.</p><p>Distance Formula (C-85)</p><p>The distance between points A(x1, y1) and B(x2, y2) is given by (AB)2 = (______)2 + (______)2 or AB = (______)2  (______)2 </p><p>Example 1: Find the distance between A(8,15) and B(-7,23).</p><p>The distance formula is also used to write the equation of a circle.</p><p>Example 2: Write an equation for the circle with center (5,4) and radius 7 units. </p><p>Let (x,y) represent any point on the ______. The distance from (x,y) to the circle’s center, (5,4) is ____. Substitute this information in the distance formula.</p><p>Example 3: Write the equation for the circle at right. </p><p>Any point on the circle = (x,y) Center = (____, ____) </p><p>Radius = ____</p><p>Example 4: Graph the quadrilateral using the given vertices. Then use the distance formula and the slope formula to determine the most specific name for quadrilateral. M(-3,4), N(-1,0), O(3,2), P(1,6) mMN =____ mNO =____ mOP =____ mPM =____ slope of MN =____ slope of NO =____ slope of OP =____ slope of PM =____</p><p>Therefore, the quadrilateral is a ______.</p><p>Work:</p><p>ÞASSIGNMENT: ______</p>