Study Guide - Geometry

Transcription

1 Study Guide - Geometry (NOTE: This does not include every topic on the outline. Take other steps to review those.) Page 1: Rigid Motions Page 3: Constructions Page 12: Angle relationships Page 14: Angle Proofs Definitions and Components of Rigid Motions Rigid Motions v taking an object and moving it to a different location without changing its shape or size Translations v shifts an object in an x or/and y direction v Components: - left/right shift - up/down shift v Notation: - T <x,y> tells you how much you move the object to the left or right or/and how much you move it down or up - for example, T <-5,2> would mean a translation of left 5 and up 2, or in other words, you would take each point and change the points to (x-5, y+2) v How to Translate an Object: - once you know how much you will be translating by in the x or/and y direction, change all of the vertices of the shape based on this - for example, if the translation is 4 right and 5 down + point A of a triangle is (2,3), you would translate A so that A would be (6, -2), then do the same thing to the other two points of the triangle v When Trying to Recognize a Translation, Look For: - the image must be congruent - the image must preserve orientation aka the lettering order - the corresponding sides of the pre-image and image are parallel unless they are on the same horizontal or vertical line as the original side Reflections v a flip around an axis of symmetry v How to Reflect a Point on a Graph: - Draw the axis of symmetry given. - Count how many units away that the point is from the axis of symmetry. - Count the same distance going over to the opposite side of the line and label the point. v When Trying to Recognize a Reflection, Look For: - If the pre-image s points are in a clockwise arrangement, the image s points must be in a counterclockwise arrangement. - If the pre-image s points are in a counter-clockwise arrangement, the image s points must be in a clockwise arrangement. Rotations v turns an object around a point v Components:

2 - center of rotation: around what? - angle of rotation: how many degrees? v Notation: - R origin, 90 would mean a rotation of 90 degrees around the origin - R 90 would also mean a rotation of 90 degrees around the origin v How to Rotate an Object: - If the problem tells you to rotate a negative angle, then the object rotates clockwise. - If the problem tells you to rotate a positive angle, then the object rotates counter-clockwise. v When Trying to Recognize a Rotation, Look For: - the letter ordering must be the same A table that is useful: Rotation of 90 : Rotation of 180 : Rotation of 270 :

3 Geometric Constructions (Part 1) Equilateral Triangle: Step 1: Start with line segment AB Step 2: Draw circle A using the radius AB Step 3: Draw circle B with radius AB Step 4: Label one intersection of the two circles Step 5: Draw line segments BC and AC to connect all 3 points All Sides of the triangle are radii of equally sized circles. Equilateral Triangle Inscribed in Circle Step 1: Draw a circle of any radius, label it as circle A Step 2: Choose a point on the border of the circle and label it as B Step 3: Draw circle B using the radius BA Step 4: Label the intersections of the circles as points C and D Step 5: Draw circle D using the radius CD Step 6: Label the intersection between circle D and circle A, E Step 7: Create line segments CD, DE and CE Equilateral Triangle Inscribed In Circle (2 nd Method) Step 1: Draw a circle of any radius, label it as circle A Step 2: Choose a point on the border of the circle and label it as B Step 3: Draw circle B using the radius BA Step 4: Label intersection as C Step 5: Draw circle C with the same radius as circle B Step 6: Label intersection as D Step 7: Repeat the process until you get back to point A Step 8: Connect every other point to get an equilateral triangle

4 Square Inscribed in a Circle Step 1: Draw circle A and mark point B anywhere to be the first vertex of the square Step 2: Draw the diameter with line AB Step 3: Name other intersection as C Step 4: Draw circle B with a radius slightly past A Step 5: Draw circle C with the same radius as circle B Step 6: Label intersections as D and E Step 7: Connect D and E to get line segment DE Step 8: Connect all 4 points to create the square. Hexagon Inscribed in a circle Steps You have to start with the given circle, with a point A in the center. (Or whatever point whey start with) 1. First, you make a point anywhere on the circle. This will be the first vertex. 2. Next, you will set the compass on point A, and set it to its radius. (The side on the circle) name the points as you make them. 3. With radius B, you will have to make an arc across the circle. 4. Next, you will move the compass to the next vertex (point C) and draw another arc. 5. Continue the steps until you have all 6 vertexes. (The length of the hexagon has to be equal to the center of the vertex) 6. Connect the points with a ruler. There should be 6 equal lines. Examples

5 As you can see above, both have 6 equal sides that are inscribed inside of a circle. The first one starts wit point a, and shows the arcs to form the hexagon. Common Mistakes Some mistakes that are normally made when trying to do this construction is not making the points equal. Many people might accidentally change the radius while making the arcs. Another mistake is connecting the points wrong. This can make them unequal swell. When you are constructing a hexagon, all the sides must be equal. Perpendicular bisectors Steps You are going to be given a line segment (AB) 1. First you place your compass on one end of the given line segment. 2. Next, you are going to set the compass to B and create circle BA 3. Then, you are going to set the compass to point A, and create circle AB 4. After you make the two circles, you are going to label the intersection points CD 5. Then you are going to draw the line segment of the two intersections. Examples

6 As you can see in the examples, here there is a line segment going across, and when the two circles you formed intersected, you create the segment. Common mistakes Some common mistakes made when creating this construction is people might create the wrong points for the circles. Another mistake is that people might not evenly crate the bisector, so it might not be even. Constructions: Part 2 - (You saw it coming) Bisecting an Angle: Angles- space between two rays Bisector- the constructed line that divides a segment or angle into two smaller, congruent pieces. 1. Begin with angle A 2. Draw circle A so it intersects both rays (any radius) 3. Label the intersections B and C 4. Draw circle B with radius BC 5. Draw circle C with radius CB 6. Label either intersection X 7. Draw segment AX. AX bisects angle Copying an Angle: 1. Begin with angle A 2. Draw ray EG (put G far away, we re not really going to use G) 3. Draw circle A of any radius and label intersections B and C 4. Draw circle E with radius AC (same radius) and label intersection as F 5. Draw circle F with radius BC 6. Label intersection of two circles as D 7. Draw ray ED A

7 Circumcenter: The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. 1. Start with triangle ABC 2. Draw perpendicular bisector of AB 3. Draw perpendicular bisector of BC 4. Draw perpendicular bisector of AC 5. Label intersection of perpendicular bisectors as O REMEMBER- ***Circumcenter involves the perpendicular bisector, not the angle bisector ***Don t forget to label the intersection of all the perpendicular bisectors, O ***Don t forget to create the circle that touches the triangle at all its vertices Incenter: The Incenter of a triangle is the point where all three angle bisectors always intersect, and is the center of the triangle's incircle. 1. Begin with triangle ABC 2. Draw angle bisector of angle A 3. Draw angle bisector of angle B

8 4. Draw angle bisector of angle C 5. Label intersection of angle bisectors as O REMEMBER- ***Incenter involves the angle bisector, not the perpendicular bisector ***Don t forget to label the intersection of all the angle bisectors, O ***Don t forget to create the circle that is inside the triangle Parallel line through a Point: 1. Begin with segment AB and point C, which is not on segment AB 2. Draw ray AC 3. Draw circle A of any radius, but not past point C. Label the two intersections D and E 4. Draw circle C with radius AD. Label the far intersection as F. 5. Draw circle F with radius DE. 6. The circles from step 4 and step 5 will intersect at two points. Label the bottom point G 7. Draw line CG.

9 REMEMBER- ***Do not pass point C when creating circle A ***The far intersection should be on ray AC ***Think of this as almost copying an angle because you want to create the same line instead of angle, on the point. So you use radius AD and then you use radius DE to create the same angles/lines (If confused by this, just ignore) Constructing an Axis of Symmetry Given Two Reflected Figures: Steps: Reflection- a flip around an axis of symmetry Changes orientation (the arrangement of the vertices in a clockwise vs. Counter-clockwise arrangement) Rigid motion Axis of Symmetry- In a reflection, the axis of symmetry is the perpendicular bisector of every line joining a point to its image 1. Choose any pre-image/image pair, and then connect them with a line. 2. Construct the perpendicular bisector of that line

10 REMEMBER- ***One pair to connect is enough Constructing a Reflected Figure Given an Axis of Symmetry: Reflection- a flip around an axis of symmetry Changes orientation (the arrangement of the vertices in a clockwise vs. Counter-clockwise arrangement) Rigid motion Axis of Symmetry- In a reflection, the axis of symmetry is the perpendicular bisector of every line joining a point to its image Steps- 1. Choose a vertex and construct a circle so it goes through two points on the axis of symmetry 2. Draw the same size circle from each of the two intersections 3. Where the two circles from step 2 intersect is the image 4. Repeat steps 1-3 for all of the other vertices

11 REMEMBER- ***Do all the vertices ***Don t forget to label them(a, B, C ) ***Do one vertice at a time instead of creating all the circle s at once ***Don t let the circle s and point s distract you!

12 Supplementary Angles: angles that have a sum of 180 Common mistakes involving supplementary angles: Often confused with complementary angles (90 ) The angles DO NOT have to be adjacent or congruent Complementary Angles: Angles that have a sum of 90 Common mistakes involving complementary angles: Often confused with supplementary angles (180 ) The angles DO NOT have to be adjacent or congruent Vertical Angles: Angles that are both opposite to one another when two lines cross and are equal Common mistakes involving vertical angles: Straight lines only! If lines are not straight, the angles aren t vertical Do not forget that they are always congruent Alternate Interior Angles: Opposite sides of the line crossing the 2 parallel lines Between the parallel lines Congruent 3 and 6 or 4 and 5 Alternate Exterior Angles: Opposite sides of the transversal Outside of the parallel lines Congruent 1 and 8 or 2 and 7 Corresponding Interior and Exterior Angles: Same side of the transversal

13 Same location Congruent 6 and 2, 5 and 1, 4 and 8, 3 and 7 Tips: Remember that alternate exterior, interior and corresponding angles are always congruent For parallel line proofs: If ( AEA, AIA, CA) are congruent, then the lines are parallel

14 Proof of Angle Relationships Using Statements and Justifications. - You need to know/memorize all the angle relationships. Here are the angle relationships and their definitions: Types of Angle Relationships 1. Supplementary Angles- two angles that form a straight angle. They add up to 180 degrees. 2. Vertical Angles- two angles that are across from each other. Vertical angles are always congruent. 3. Complementary Angles- two angles that form a right angle, which add up to 90 degrees. 4. Corresponding Angles- Two parallel lines are crossed by another line (called the Transversal), the angles in matching corners are called corresponding angles which are always equal. 5. Alternate Interior Angles- two angles inside of the parallel lines that are on different sides of the transversal. they always have the same measure. 6. Alternate Exterior Angles- two angles outside of the parallel lines that are on different sides of the transversal. they always have the same measure. 4, 5, and 6: ALL THREE ANGLES ARE CONGRUENT! - Different types of justifications include: 1. definitions (ex: angles) 2. properties (ex: all are congruent) 3. conditionals (ex: if <something is true>, then <something else is true> - Proving angle congruence requires you to show or explain how two angles equal each other. To do this you need to make a chart. One side of the chart would have your statements and on the opposite side you would write your justifications about the statement you just made. - FOR EXAMPLE: Let s try solving this problem together.

15 Question: What does m equal, and how do you know? Prove it by using a two column chart. How to Solve: Since vertical angle are congruent, we can make 2m+10= 5m- 80. Then we just solve this equation. We can subtract 2m from both sides and add 80 on both sides. This will make the equation 90=3m. Then divide 3 into 90. That will give you a answer of m= 30. Now the first part of the question is done. The next part of the question is very important. The chart you will need to make will look something like this : Of course ours is going to look a little bit different, but

16 you get the idea. First you want to state what s given. In this case that would be that line CD intersects BA. then for the justification you would write given since it s right there for you. Next you would write angle CEA is equal to angle BED. For the justification you would write vertical angles are always congruent. Then the next thing you would write is 2m+10=5m- 80. In this step you are just plugging in the equations give for angle CEA and BED. Last but not least you would solve it and justify it with algebra. Common Mistakes: Many people tend to forget about writing angles. You cannot just state the brief rule. Also, it is important to state that there is a transversal. Important Tips: Look for the angle relationships and use the definition as the justification.