Geometric and Algebraic Connections Geometric and Algebraic Connections Triangles, circles, rectangles, squares... We see shapes every day, but do we know much about them?? What characteristics do they have and how do you know? While we can say many things about shapes, there are proven statements (called theorems) that deal with the characteristics of these shapes. In order to gain a better understanding of the properties and characteristics of these shapes, we will look at these shapes on a coordinate plane just like we do linear equations. Prepare to be amazed at all of things you will discover about the points, lines, and polygons. At this point we know how to graph linear, exponential, and quadratic equations. How can we graph more interesting figures? What about a circle? It is possible to start with the math that we have learned so far and derive the equation of a circle and even an ellipse! Think of the pictures that you can create with all of those shapes. The possibilities are endless! Essential Questions How can we determine if two lines are parallel, perpendicular, or neither? What are simple ways to consider and calculate the perimeters and areas of polygons? When finding the distance between two points, what simple shape can we make, and how can these help us find the distance between these two points? How can you use the Pythagorean Theorem to derive the equation of a circle? How can coordinates be used to prove geometric theorems? Module Minute Parallel lines never intersect. Lines are parallel if they have the same slope. Because they have the same slope, the lines will never intersect making them parallel. Finding the perimeter of polygons simply involves adding the lengths of all of its sides. Finding the area of polygons involving using formulas (if applicable) or considering the number of square units the polygon contains. While this can be somewhat difficult, we can divide a polygon into smaller polygons in order to find the area much easily. When finding the distance between two points, we can use the distance formula, or we can create a right triangle between the two points with a vertical and horizontal segment between the two points and simply use the Pythagorean Theorem we already know. What to Expect In this unit you will be responsible for completing the following assignments. Converting Standard Form to General Form Assignment Circles and Parabolas Quiz Systems of Equations Discussion Dr. Cone's New House Project Geometric and Algebraic Connections Test Key Terms Distance The amount of space or separation between two points. Parallel Lines Lines that never intersect due to the lines having the same slope. Perpendicular Lines Lines that intersect and whose intersection forms right angles due to the lines having slopes that are opposite reciprocals Slope Rate of change or rise over run between two points. Theorem A mathematical statement accepted as a true statement.
Center of a Circle The point inside the circle that is the same distance from all of the points on the circle. Circle The set of all points in a plane that are the same distance, called the radius, from a given point, called the center. Diameter The distance across a circle through its center. General Form of a Circle Pythagorean Theorem A theorem that states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs. Radius The distance from the center of a circle to any point on the circle. Standard Form of a Circle Pythagoras and the Pythagoreans, where (h, k) is the center and r is the radius. More than 2,500 years ago, around 530 BCE, a man by the name of Pythagoras founded a school in modern southeast Italy. Members of the school, which was actually more of a brotherhood, were bound by a pledge of allegiance to their master Pythagoras and took an oath of silence to not divulge secret discoveries. Pythagoreans shared a common belief in the supremacy of numbers, using them to describe and understand everything from music to the physical universe. Studying a wide range of intellectual disciplines, Pythagoreans made a multitude of discoveries, many of which were attributed to Pythagoras himself. No records remain of the actual discoverer, so the identity of the true discoverer may never be known. Perhaps the most famous of the Pythagoreans' contributions to knowledge is proving what has come to be known as the Pythagorean Theorem. Pythagorean Theorem The Pythagorean Theorem allows you to find the lengths of the sides of a right triangle, which is a triangle with one 90 angle (known as the right angle). An example of a right triangle is depicted below. A right triangle is composed of three sides: two legs, which are labeled in the diagram as, and a hypotenuse, which is the side opposite to the right angle. The hypotenuse is always the longest of the three sides. Typically, we denote the right angle with a small square, as shown above, but this is not required. The Pythagorean Theorem states that the length of the hypotenuse squared equals the sum of the squares of the two legs. This is written mathematically as : To verify this statement, first explicitly expressed by Pythagoreans so many years ago, let's look at an example. Example 1 Consider the right triangle below. Does the Pythagorean Theorem hold for this triangle?
Solution As labeled, this right triangle has sides with lengths 3, 4, and 5. The side with length 5, the longest side, is the hypotenuse because it is opposite to the right angle. Let's say the side of length 4 is and the side of length 3 is. Recall that the Pythagorean Theorem states: If we plug the values for the side lengths of this right triangle into the mathematical expression of the Pythagorean Theorem, we can verify that the theorem holds: (4)² + (3)² = (5)² 16 + 9 = 25 25 = 25 Although it is clear that the theorem holds for this specific triangle, we have not yet proved that the theorem will hold for all right triangles. A simple proof, however, will demonstrate that the Pythagorean Theorem is universally valid. Proof Based on Similar Triangles The diagram below depicts a large right triangle (triangle ABC) with an altitude (labeled h) drawn from one of its vertices. An altitude is a line drawn from a vertex to the side opposite it, intersecting the side perpendicularly and forming a 90 angle. In this example, the altitude hits side AB at point D and creates two smaller right triangles within the larger right triangle. In this case, triangle ABC is similar to triangles CBD and ACD. When a triangle is similar to another triangle, corresponding sides are proportional in lengths and corresponding angles are equal. In other words, in a set of similar triangles, one triangle is simply an enlarged version of the other. Similar triangles are often used in proving the Pythagorean Theorem, as they will be in this proof. In this proof, we will first compare similar triangles ABC and CBD, then triangles ABC and ACD. Comparing Triangles ABC and CBD
In the diagram above, side AB corresponds to side CB. Similarly, side BC corresponds to side BD, and side CA corresponds to side DC. It is possible to tell which side corresponds to the appropriate side on a similar triangle by using angles; for example, corresponding sides AB and CB are both opposite a right angle. Because corresponding sides are proportional and have the same ratio, we can set the ratios of their lengths equal to one another. For example, the ratio of side AB to side BC in triangle ABC is equal to the ratio of side CB to corresponding side BD in triangle CBD: Written with variables, this becomes: Next, we can simplify this equation by multiplying both sides of the equation by the common denominator ax: With simplification, we obtain: cx = a² Comparing Triangles ABC and ACD
Triangle ABC is also similar to triangle ACD. Side AB corresponds to side CA, side BC corresponds to side CD, and side AC corresponds to side DA. Using this set of similar triangles, we can say that: Written with variables, this becomes: Similar to before, we can multiply both sides of the equation by and : Earlier, we found that cx = a². If we replace cx with a², we obtain c² = a² + b². This is just another way to express the Pythagorean Theorem. In the triangle ABC, side c is the hypotenuse, while sides a and b are the two legs of the triangle. The Theorem's Significance Although the Babylonians may be the first to understand the concepts of the Pythagorean Theorem and the Pythagoreans were the first to explicitly prove it, Euclid of Alexandria, active around 300 BCE, was the man responsible for popularizing the theorem. Euclid, head of the department of mathematics at a school in Alexandria, took it upon himself to compile all knowledge about mathematics known at his point in history. The result was a book called Elements, which included two of Euclid's own proofs of the Pythagorean Theorem. The propagation of this theorem is significant because the theorem is applicable to a variety of fields and situations. Though the theorem is fundamentally geometric, it is useful in many branches of science and mathematics, and you are likely to encounter it often as you continue to study more advanced topics. The Pythagorean Theorem, however, is also relevant to a variety of situations in everyday life. Architecture, for instance, employs the concepts behind the Pythagorean Theorem. Measuring and computing distances will also often involve using this theorem. Televisions, when advertised, are measured diagonally; for example, a television may be listed as ''a 40 inch," meaning that its diagonal is 40 inches long. The length of the television, the width of the television, and the Pythagorean Theorem were used to get this measurement.
Look out for ways that you can use this theorem in your everyday life there may be more than you expected! The Pythagorean Theorem is useful in a variety of mathematical situations because it can be applied to solve many different types of problems. As we have seen, the most basic application of the theorem is finding the length of one side of a right triangle when the lengths of the other two sides are known. In this chapter, we'll expand on other applications of the theorem. Pythagorean Triples As mentioned earlier in the ''History of the Pythagorean Theorem" chapter, the Babylonians demonstrated an understanding of the Pythagorean Theorem by listing Pythagorean triples on a clay tablet. Pythagorean triples are sets of three integers positive whole numbers that make a right triangle. Pythagorean triples are frequently used in examples and problems, making it worthwhile to memorize some of the more common triples. Pythagorean triples are frequently used in examples and problems, making it worth while to memorize some of the more common triples. The most common Pythagorean triples are (3, 4, 5) and (5, 12, 13). Multiples of these triples such as (6, 8, 10) are also Pythagorean triples. Pythagorean triples are frequently used in examples and problems, making it worthwhile to memorize some of the more common triples. Though these triples are the most common, there is an infinite number of combinations of integers that satisfy the Pythagorean Theorem. Table below lists all primitive triples with a hypotenuse length less than 100. Note that the set (6, 8, 10) is not listed in the table because it is a multiple of the primitive triple (3, 4, 5). Content from this page found at www.ck12.org and khanacademy.org CCBY Finding Distances on a Coordinate Grid
Finding Distances on a Coordinate Grid You can also apply the Pythagorean Theorem to find the distance between two points on a coordinate grid. Let's look at several examples to see how this is done. Example 1 Find the distance between the two points on the coordinate system below. Solution The line segment connecting these two points is neither horizontal nor vertical, so it is not possible to simply count the number of spaces on the grid. It is, however, possible to think of the line segment connecting the two points as the hypotenuse of a right triangle. Drawing a vertical line at x = 2 and a horizontal line at y = 1 will create a right triangle, as shown in the figure below. It is easy to determine the lengths of the legs of this right triangle because they run parallel to the x and y axes. You can simply count on the grid how long each leg is. The horizontal leg has a length of 4 and the vertical leg also has a length of 4. Using the Pythagorean Theorem, you can find the length of the diagonal line, which is also the hypotenuse of the right triangle with two legs of length 4.
The line segment connecting the points ( 2, 1)and (2, 3) is about 5.657 units long. The Distance Formula We can generalize the process used in the previous example to apply to any situation where you want to find the distance between two coordinates. Using points ( ) and ( ), we can derive a general distance formula. Similar to the previous example, we will let the line segment connecting two coordinates be the hypotenuse of a right triangle. Let's start by finding the length of the horizontal leg by finding the difference in the x coordinates. The difference in the x coordinates would be. The absolute value brackets are used to indicate that the length of the horizontal leg must be a positive value because a negative distance does not have any physical meaning. We can also find the length of the vertical leg by finding the difference in y coordinates. The difference in the y coordinates would be value brackets are used because lengths cannot be negative.. Once again, absolute Now that we have found the lengths of the legs of the right triangle we have created, we can plug them into our equation for the Pythagorean Theorem. At this point, it is not necessary to use the absolute value brackets because any value squared will be positive. For example, even if were a negative value, it would become a positive value when squared. Therefore, we can rewrite the equation above without the absolute value brackets, and the expression will remain the same: To solve for the distance, we can take the roots of both sides and obtain the distance formula: To check this formula with an example, lets plug in the coordinates from the previous example and see if we get the same distance as the answer. The coordinates from the previous example were ( 2, 1) and (2,3), and we determined that the distance between these two points was about 5.657 units. Though the distance formula is different from the Pythagorean Theorem, the formula is simply another form of the theorem and is not an entirely different concept. It is not necessary to memorize this equation because you can simply think about how the Pythagorean Theorem applies to a line segment on a coordinate system. Nonetheless, it is helpful to know that this equation holds for any two coordinates. Content from this page found at www.ck12.org and khanacademy.org CCBY Parallel and Perpendicular Line Postulates
Parallel Line Postulate Parallel Postulate: For any line and a point not on the line, there is one line parallel to this line through the point. There are infinitely many lines that go through A, but only one that is parallel to l. Perpendicular Line Postulate Perpendicular Line Postulate: For any line and a point not on the line, there is one line perpendicular to this line passing through the point. There are infinitely many lines that pass through A, but only one that is perpendicular to l. Angles and Transversals
Transversal: A line that intersects two other lines. The area between l and m is the interior. The area outside l and m is the exterior. Looking at t; l and m, there are 8 angles formed. They are labeled below. There are 8 linear pairs and 4 vertical angle pairs. An example of a linear pair would be 1 and 2. An example of vertical angles would be 5 and 8. Example 1: List all the other linear pairs and vertical angle pairs in the picture above. Solution: Linear Pairs: 2 and 4; 3 and 4; 1 and 3; 5 and 6; 6 and 8; 7 and 8; 5 and 7 Vertical Angles: 1 and 4; 2 and 3; 6 and 7. There are also 4 new angle relationships. Corresponding Angles: Two angles that are on the same side of the transversal and the two different lines. Imagine sliding the four angles formed with line l down to line m. The angles which match up are corresponding.
Above, 2 and 6 are corresponding angles. Alternate Interior Angles: Two angles that are on the interior of l and m, but on opposite sides of the transversal. Above, 3 and 5 are alternate exterior angles. Alternate Exterior Angles: Two angles that are on the exterior of l and m, but on opposite sides of the transversal. Above, 2 and 7 are alternate exterior angles. Same Side Interior Angles: Two angles that are on the same side of the transversal and on the interior of the two lines.
Above, 3 and 5 are same side interior angles. Content from this page found at www.ck12.org and khanacademy.org CCBY Parallel and Perpendicular Lines in the Coordinate Plane Know What? The picture to the right is the California Incline, a short road that connects Highway 1 with Santa Monica. The length of the road is 1532 feet and has an elevation of 177 feet. You may assume that the base of this incline is zero feet. Can you find the slope of the California Incline? HINT: You will need to use the Pythagorean Theorem! Slope in the Coordinate Plane Watch the videos in the video showcase below. You can move from one video to the next by using the arrow in the upper right hand corner. Recall from earlier lessons, two points have a slope of. Flip through the album below to learn the Types of Slope. Slopes of Parallel Lines Earlier in the chapter we defined parallel lines as two lines that never intersect. In the coordinate plane, that would look like this:
If we take a closer look at these two lines, the slopes are both 2/3. This can be generalized to any pair of parallel lines. Parallel lines have the same slope. Example 1 Find the equation of the line that is parallel to y = 1/3 x + 4 and passes through (9, 5). Solution: Recall that the equation of a line is y = mx + b, where m is the slope and b is the y intercept. We know that parallel lines have the same slope, so the line will have a slope of 1/3. Now, we need to find the y intercept. Plug in 9 for x and 5 for y to solve for the new y intercept (b). The equation of line is y= 1/3x 2.. Parallel lines always have the same slope and different y intercepts. Slopes of Perpendicular Line Perpendicular lines are two lines that intersect at a 90, or right, angle. In the coordinate plane, that would look like this: If we take a closer look at these two lines, the slope of one is 4 and the other is 1/ 4. This can be generalized to any pair of perpendicular lines in the coordinate plane. The slopes of perpendicular lines are opposite signs and reciprocals of each other. Find the slope of the perpendicular lines to the lines below. Example 2 Find the equation of the line that is perpendicular to y = 1/3 x + 4 and passes through (9, 5). Solution
First, the slope is the reciprocal and opposite sign of 1/3. So, m = 3. Plug in 9 for x and 5 for y to solve for the new y intercept (b). 5 = 3(9) + b 5 = 27 + b 32 = b Therefore, the equation of line is y = 3x 32. Content from this page found at www.ck12.org and khanacademy.org CCBY Graphing Parallel and Perpendicular Lines Example 1 Find the equation of the lines below and determine if they are parallel, perpendicular or neither. Solution: The top line has a y intercept of 1. From there, use ''rise over run" to find the slope. From the y intercept, if you go up 1 and over 2, you hit the line again, m = 1/2. The equation is y = 1/2 x + 1. For the second line, the y intercept is 3. The ''rise" is 1 and the ''run" is 2 making the slope 1/2. The equation of this line is y = 1/2 x 3. The lines are parallel because they have the same slope. Example 2 Graph 3x 4y = 8 and 4x + 3y = 15. Determine if they are parallel, perpendicular, or neither. Solution First, we have to change each equation into slope intercept form. In other words, we need to solve each equation for y. 3x 4y = 8 4y = 3x + 8 y = 3/4x 2 4x + 3y = 15
3y = 4y + 15 y = 4/3x + 5 Now that the lines are in slope intercept form (also called y intercept form), we can tell they are perpendicular because the slopes are opposites signs and reciprocals. Example 3 Find the equation of the line that is (a) parallel to the line through the point. (b) perpendicular to the line through the points. Solution First the equation of the line is y = 2x+6 and the point is (2, 2). The parallel would have the same slope and pass through (2, 2). y = 2x + b 2 = 2(2) + b 2 = 4 + b 6 = b The equation is y = 2x + 6
The perpendicular line also goes through (2, 2), but the slope is 1/2 y = 1/2x + b 2 = 1/2(2) + b 2 = 1 + b 1 = b The equation is y = 1/2 x 1 Know What? Revisited In order to find the slope, we need to first find the horizontal distance in the triangle to the right. This triangle represents the incline and the elevation. To find the horizontal distance, we need to use the Pythagorean Theorem, a² + b² = c², where c is the hypotenuse. Content from this page found at www.ck12.org and khanacademy.org CCBY The Distance Formula Know What? The shortest distance between two points is a straight line. Below are distances between cities in the Los Angeles area. What is the longest distance between Los Angeles and Orange? Which distance is the shortest? The distance between two points ( ) and ( ) can be defined as: Example 1
Find the distance between (4, 2) and ( 10, 3). Solution Plug in (4, 2) for ( ) and ( 10, 3) for ( ) and simplify. Example 2 Find the distance between ( 2, 3) and (3, 9). Solution Use the distance formula, plug in the points, and simplify. Distances are always positive! Shortest Distance between Vertical and Horizontal Lines All vertical lines are in the form x = a, where a is the x intercept. To find the distance between two vertical lines, count the squares between the two lines. Example 1 Find the distance between x = 3 and x = 5.
Solution: The two lines are 3 ( 5) units apart, or 8 units apart. You can use this method for horizontal lines as well. All horizontal lines are in the form y = b, where b is the y intercept. Example 2 Find the distance between y = 5 and y = 8. Solution: The two lines are 5 ( 8) units apart, or 13 units. Shortest Distance between Parallel Lines with m = 1 or 1 The shortest distance between two parallel lines is the perpendicular line between them. There are infinitely many perpendicular lines between two parallel lines. Notice that all of the pink segments are the same length.
Example 3 Find the distance between y = x + 6 and y = x 2. Solution: 1. Find the perpendicular slope. m = 1, so = 1 2. Find the y intercept of the top line, y = x + 6. (0, 6) 3. Use the slope and count down 1 and to the right 1 until you hit y = x 2. Always rise/run the same amount for m = 1 or 1.
4. Use these two points in the distance formula to determine how far apart the lines are. Content from this page found at www.ck12.org and khanacademy.org CCBY