Geometry - Coordinate Geometry - Quick Guides

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Study GuideGeometry–Coordinate Geometry1. Distance Formula1.1Understanding the Distance FormulaFigure 1TheDistance Formulahelps us find the straight-line distance between two points on a coordinateplane. To use this formula, you’ll need the coordinates of the two points. For example, let's look atthree points: A, B, and C.In the first image, we see the points:•A at (2, 2)•B at (5, 2)•C at (5, 6)The goal is to calculate the distance between point A and point C, which we’ll callAC.

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Study Guide1.2How Do We Calculate the Distance?To findAC, we can't just subtract the x-values and y-values directly, because it's not a straighthorizontal or vertical line. Instead, triangle ABC forms a right triangle withACbeing the hypotenuse.We use thePythagorean Theoremhere, which states that the square of the length of thehypotenuse (AC) equals the sum of the squares of the lengths of the other two sides (AB and BC).So, we start by calculating AB and BC.Step 1: Find AB and BC•AB= |5-2| = 3•BC= |6-2| = 4Step 2: Apply the Pythagorean TheoremWe can now findACusing:So,AC = 5 units.1.3The Distance FormulaThis process follows the general formula for finding the distance between any two points ((x1, y1)) and((x2, y2)):This formula tells us how to calculate the distancedbetween two points using their coordinates.

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Study GuideExample1:Finding Distance Between Two PointsLet's practice with another example. Here are two points with the following coordinates:•Point 1: (-3, 4)•Point 2: (5, 2)We want to find the distance between them. Using the Distance Formula:So, the distance between these two points is approximately2√17.Example2:Showing an Isosceles TriangleHere’s an example of how to use the Distance Formula to show that a triangle isisosceles. We havea triangle with the following points:•Point A: (12, 5)•Point B: (5, 3)•Point C: (12, 1)To check if the triangle is isosceles, we need to show that two sides are equal in length. We use theDistance Formula to calculateABandBC.Step 1: Find AB and BCSinceAB = BC, the triangle isisosceles.

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Study GuideSummary•TheDistance Formulahelps us find the distance between two points on a coordinate plane.•The formula is:•Wecan use it to find distances, verify triangle properties, and solve many geometricproblems.2. Midpoint FormulaThe midpoint of a line segment is simply the point that is exactly halfway between its two endpoints.We can think of it as the "average" of the coordinates of the two points.To find the midpoint of a line segment, we use the midpoint formula:If the coordinates of the two endpoints are((x1, y1))and((x2, y2)), then the midpoint(M)is given by thefollowing formula:This means, to get the(x)-coordinate of the midpoint, you average the(x)-coordinates of theendpoints, and do the same for the(y)-coordinates.Example1:Finding the MidpointLet's look at an example to understand how to use the formula in action.

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Study GuideFigure 1Suppose we have two points:(Q(-9,-1))and(T(-3, 7)), and we need to find the midpoint(R)betweenthem.Using the midpoint formula:So, the midpoint(R)is at the coordinates((-6, 3)).To check if(R)is truly the midpoint, we can use the distance formula to make sure that the distancesfrom(R)to(Q)and from(R)to(T)are the same.Using the distance formula:

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Study GuideSubstitute the coordinates into the formula:The distances match, confirming that(R)is indeed the midpoint!Example2:Finding the Missing EndpointWhat if we know the midpoint and one endpoint, and we need to find the other endpoint?Let's say the midpoint(A(-3, 8))and the endpoint(I(12,-1))are given. We want to find the coordinatesof point(B).The midpoint formula tells us:We know that the midpoint's coordinates are((-3, 8)), so we set up two equations:Multiply both sides of each equation by 2:Solving these equations gives us:So, the coordinates of point(B)are((-18, 17)).

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Study GuideSummary1.The midpoint formula finds the point exactly halfway between two endpoints.2.To verify if the midpoint is correct, use the distance formula to compare distances.3.The distance from the midpoint to each endpoint should be the same.4.If you know the midpoint and one endpoint, you can find the other endpoint.5.Apply the midpoint formula in different scenarios for practice and understanding.3. Slope of a LineThe slope of a line is a way to measure how steep the line is and in which direction it moves. It showshow much the line rises or falls as it moves from left to right.•When a line rises from left to right, the slope ispositive.•When a line falls from left to right, the slope isnegative.•A horizontal line has aslope of 0.•A vertical line has anundefined slope.3.1Different SlopesLook at the examples below to understand how different lines have different slopes:•Figure 1(a): A line with apositive sloperises as you move from left to right.•Figure 1(b): A line with anegative slopefalls as you move from left to right.•Figure 1(c): Ahorizontal linehas a slope of zero.•Figure 1(d): Avertical linehas an undefined slope.

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Study GuideFigure 13.2The Slope FormulaTo find the slope between two points, we use this formula:Where(m)is the slope, and((x1, y1))and((x2, y2))are the coordinates of the two points.•If the line is vertical, then the slope is undefined because we would be dividing by 0 (since(x1= x2)).

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Study GuideExample1:Finding the Slope of a LineFigure2Let’s practice using the slope formula. Look at the diagram below, where we have four different lineslabeled a, b, c, and d. We will calculate their slopes.•(a)Line a passes through the points((-7, 2))and((-3, 4)). To find the slope, use the formula:•(b)Line b passes through the points((2, 4))and((6,-2)). Using the formula:•(c)Line c ishorizontal, so its slope is(m = 0).•(d)Line d isvertical, so its slope isundefined.

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