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Geometry Honors

Geometry Honors. Day 4. Today’s Objectives. Linear Measurement Distance Formula Midpoint Angle Measure Assignment. Line Segment. A line segment is a set of points consisting of two points on a line (‘endpoints’) and all the points in between.

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Geometry Honors

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  1. Geometry Honors Day 4

  2. Today’s Objectives • Linear Measurement • Distance Formula • Midpoint • Angle Measure • Assignment

  3. Line Segment • A line segment is a set of points consisting of two points on a line (‘endpoints’) and all the points in between. • A line segment is named after its two endpoints, with a bar over them. • Ex: • Note: would refer to the same line segment. B A

  4. Ray • A ray is a set of points consisting of a point on a line and all the points on one side of that point. • A ray continues indefinitely in one direction • A ray is named after its endpoint and a second point on the ray, with an arrow over them. • Ex: • Note: would NOT refer to the same ray. B A

  5. Length and Congruence • The length of a line segment is the distance between its two endpoints. When we refer to the length of a line segment, we don’t use the overhead bar. • Congruent figures have the same shape and size. The symbol for congruency is . • When marking diagrams, ‘hash marks’ indicate congruency. The number of hash marks signify which segments are congruent to which. A B D E C

  6. Congruence and Equality • Note: We use equal when talking about the measurement of figures, and congruent when talking about the figures themselves. • This segment has a length of five units • The lengths of these segments are equal • These two segments are congruent

  7. Congruence and Equality • Note: In this course, we do not rely on measurements to determine congruency. We do not assume that figures are drawn to scale. • If two figures are marked congruent, then they are congruent no matter what they look like.

  8. Betweenness of points • Any two real numbers have an infinite amount of real numbers between them. • Likewise, any two points on a line have an infinite amount of points in between them. • Point A is between points C and T if and only if C, A, and T are collinear and CA + AT = CT C A T

  9. Adding Segments • We can use this concept to add and subtract the lengths of line segments. • Find CT if CA = 7 and AT = 21. • Find CA if CT = 5x + 4 and AT = 2x – 3. C A T

  10. Distance • The distance between two points is the length of the segment with those two points as its endpoints. • The distance between two points on a number line is the absolute value of the difference between their coordinates. • If points A and B have coordinates of x1 and x2, respectively, then AB = │x1 – x2│ A B x1 x2

  11. Example • Find the following lengths: • AB • BD • AD A B C D -4 1 8 12

  12. Opposite rays • If two rays share an end point, and point in the opposite directions, they are opposite rays.

  13. ANGLE A geometric figure formed by two rays with a common endpoint (the vertex). A A B BAC D CAB C

  14. ANGLE The measure of an angle describes how far the rays have been spread apart. The units of measurement are degrees.

  15. CLASSIFICATION OF ANGLES • Acute: Measurement is less than 90 • Right: Measurement is 90 • Obtuse: Measurement is between 90 and 180 • Straight: Measurement is 180

  16. Bisecting an Angle • A line, ray, or line segment that bisects an angle is called an angle bisector. • An angle bisector creates two smaller, congruent angles that are each half the measure of the original.

  17. Triangles • A triangle is a figure formed by three line segments (sides) connected at their endpoints in a closed circuit. The points where two sides meet are called vertices.

  18. Triangles • One way to classify triangles is by their angles. • An acute triangle only has acute angles. • A right triangle has one right angle. • An obtuse triangle has one obtuse angle.

  19. Right Triangles • The sides of a right triangle have special names. • The longest side (always across from the right angle) is the hypotenuse. • The other two sides are the legs. leg leg hypotenuse

  20. The Pythagorean Theorem • The sum of the areas of the squares whose sides are the legs of a right triangle is equal to the area of the square whose side is the hypotenuse of that triangle. • This relationship among the sides of right triangles is known as the Pythagorean Theorem in this culture. (Proof will come later.) • In simplest form a2 + b2 = c2, where a and b are the lengths of the legs and c is the length of the hypotenuse.

  21. The Pythagorean Theorem and Length • The Pythagorean Theorem is one of the most frequently used ideas in this course. • Most problems involving finding length or distance will involve the Pythagorean Theorem at some point. • On your graph paper, draw a pair of axes and plot (1, 5) and (4, 1). Connect them to form a line segment. • How can we find the length of this segment without measuring? • Can we generalize this process to find the distance between any two points (x1, y1) and (x2, y2)?

  22. The Distance Formula • The distance formula is the Pythagorean Theorem rewritten to make it convenient to plug in coordinates.

  23. Midpoint and Bisection • A midpoint is the point that divides a line segment into two smaller, congruent segments. Each smaller segment is half the length of the original. Y X Z

  24. Midpoint and Bisection • To bisect means to divide into two congruent pieces. The midpoint of a line segment bisects that segment. • A segment can be bisected by a point, line, ray, or another segment. P X Z

  25. More with Midpoints (Cartesian Analysis) • On graph paper, draw a single number line, label zero in the center, and mark the points 5 and -3. • What are some ways to find the midpoint of this line segment? • Now draw a pair of axes, and mark (6, 2) and (1, -4). Measure the length of this segment and estimate where the midpoint is. • How can we find this without measuring?

  26. Midpoint Formula • The midpoint formula is:

  27. Homework #2 • Workbook pp. 4, 6, 8 • Bring compass and straightedge next class!

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