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Geometry - Coordinate Geometry

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    Study Guide Geometry – Coordinate Geometry 1. Distance Formula 1.1 Understanding the Distance Formula Figure 1 The Distance Formula helps us find the straight - line distance between two points on a coordinate plane. To use this formula, you’ll need the coordinates of the two points. For example, let's look at three 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 call AC .

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    Study Guide 1.2 How Do We Calculate the Distance? To find AC , we can't just subtract the x - values and y - values directly, because it's not a straight horizontal or vertical line. Instead, triangle ABC forms a right triangle with AC being the hypotenuse. We use the Pythagorean Theorem here, which states that the square of the length of the hypotenuse (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| = 4 Step 2: Apply the Pythagorean Theorem We can now find AC using: So, AC = 5 units . 1.3 The Distance Formula This process follows the general formula for finding the distance between any two points ((x 1 , y 1 )) and ((x 2 , y 2 )): This formula tells us how to calculate the distance d between two points using their coordinates.

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    Study Guide Example 1 : Finding Distance Between Two Points Let'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 approximately 2√17 . Example 2 : Showing an Isosceles Triangle Here’s an example of how to use the Distance Formula to show that a triangle is isosceles . We have a 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 the Distance Formula to calculate AB and BC . Step 1: Find AB and BC Since AB = BC , the triangle is isosceles .

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    Study Guide Summary • The Distance Formula helps us find the distance between two points on a coordinate plane. • The formula is: • We can use it to find distances, verify triangle properties, and solve many geometric problems. 2. Midpoint Formula The 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 ( (x 1 , y 1 ) ) and ( (x 2 , y 2 ) ) , then the midpoint ( M ) is given by the following formula: This means, to get the ( x ) - coordinate of the midpoint, you average the ( x ) - coordinates of the endpoints, and do the same for the ( y ) - coordinates. Example 1 : Finding the Midpoint Let's look at an example to understand how to use the formula in action.

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    Study Guide Figure 1 Suppose we have two points: ( Q( - 9, - 1) ) and ( T( - 3, 7) ) , and we need to find the midpoint ( R ) between them. 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 distances from ( R ) to ( Q ) and from ( R ) to ( T ) are the same. Using the distance formula:

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    Study Guide Substitute the coordinates into the formula: The distances match, confirming that ( R ) is indeed the midpoint! Example 2 : Finding the Missing Endpoint What 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 coordinates of 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 Guide Summary 1. 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 Line The slope of a line is a way to measure how steep the line is and in which direction it moves. It shows how much the line rises or falls as it moves from left to right. • When a line rises from left to right, the slope is positive . • When a line falls from left to right, the slope is negative . • A horizontal line has a slope of 0 . • A vertical line has an undefined slope . 3.1 Different Slopes Look at the examples below to understand how different lines have different slopes: • Figure 1(a) : A line with a positive slope rises as you move from left to right. • Figure 1(b) : A line with a negative slope falls as you move from left to right. • Figure 1(c) : A horizontal line has a slope of zero. • Figure 1(d) : A vertical line has an undefined slope.

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    Study Guide Figure 1 3.2 The Slope Formula To find the slope between two points, we use this formula: Where ( m ) is the slope, and ( (x 1 , y 1 ) ) and ( (x 2 , y 2 ) ) 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 ( x 1 = x 2 ) ).

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    Study Guide Example 1 : Finding the Slope of a Line Figure 2 Let’s practice using the slope formula. Look at the diagram below, where we have four different lines labeled 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 is horizontal , so its slope is ( m = 0 ) . • (d) Line d is vertical , so its slope is undefined .

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