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Chapter 7: Right Triangles and Trigonometry You have: 2 sides of a right triangle You need: Examples: Use: Pythagorean Theorem: a b c *Remember c is always the hypotenuse (longest side) Converse of Pythagorean Theorem: *Be sure to check to see if a triangle exists FIRST!!* 2 the last side 3 sides of a triangle To determine if right, acute or obtuse a 45˚-45˚-90˚ triangle and one side another side 2 2 Right Triangle: c 2 a 2 b 2 Acute Triangle: c 2 a 2 b 2 Obtuse Triangle: c 2 a 2 b 2 hypotenuse=leg 2 Figure out if you need a leg or a hypotenuse and plug it in! *Remember if you have a leg, and you are looking for a leg - they are equal in an isosceles triangle! hypotenuse= 2(short leg) long leg= short leg 3 a 30˚-60˚-90˚ triangle and one side the last two sides one angle and one side and it is NOT a special right triangle another side Trig Ratios (SOH CAH TOA) Label your parts of the triangle with your Opposite, Adjacent, Hypo. Figure out which trig ratio you are using and set up your equation. 2 sides an angle Inverse Trig Ratios Look at what angle you want to find and label your sides with your Opposite, Adjacent, Hypo accordingly. Set up your trig ratio then multiply each side by its inverse. Figure out which side is which by looking at the angles (remember short leg is across from the 30˚ angle, long leg is across from the 60˚). Once you know which one you are starting with, pick an appropriate equation to plug it into. Rectangle Chapter 8: Quadrilaterals All angles are right angles Diagonals are congruent Parallelograms Quadrilaterals 4 sides Interior angles add up to 360˚ Exterior angles add up to 360˚ 2 pairs of parallel lines Opposite sides congruent and parallel Opposite angles congruent Diagonals bisect each other Consecutive angles are supplementary 1 pair of parallel lines Trapezoid Rhombus 4 sides are equal Diagonals are perpendicular Diagonals bisect the angles Isosceles Trapezoid 2 pairs of congruent base angles Legs are congruent Diagonals are congruent 0 pairs of parallel lines Kite x, y x, y across y-axis: x, y x, y across y=x: x, y y, x across y=-x x, y y, x 90˚ x, y y, x 180˚ x, y x, y 270˚ x, y y, x Finding Segment Lengths Polygon Formulas: Sum of interior ’s: (n 2)180 ONE interior of regular polygon : ( n 2)180 n 2 pairs consecutive congruent sides Diagonals are perpendicular 1 pair of congruent angles Non-congruent angles are bisected by the diagonal Chapter 9: Properties of Transformations translation (slide): x, y x h, y k Square Sum of exterior ’s: 360 ONE exterior of regular polygon: 360 n *n: number of sides Chapter 10: Properties of Circles reflection (flip): across x-axis: rotation (turn): Tangent segments from the same point are congruent: AB CB Secant segments Secant and Tangent BC AC DC EC AB 2 CB DB ow=ow Chords AE EB DE EC Finding Angle Measures glide reflection: translate, then reflect tessellation- figures that cover a plane with no gaps or overlap {square, triangle, hexagon} Symmetry: Line: -figure has at least one line of reflection; folds exactly in half, one half on top of the other mRPS mRS Point(Rotational): can rotate the figure 180° and it looks the same, matches onto itself Angle ON circle Angle IN circle Finding arc length Set up a proportion: measure of arc = length of arc 360˚ circumference Angle OUSTIDE circle Angle at CENTER Finding area of a sector Set up a proportion: measure of arc= area of the sector 360˚ area of the circle Chapter 11 – 12: Measuring Length and Area ~ Surface Area and Volume – FORMULA SHEET 2 3 *Know Lateral Area (NO bases – just faces) vs. Scale Factor: a ~ Ratio of Areas: a ~ Ratio of Volumes: a 2 3 Surface Area (faces & bases) vs. Volume (FILL) b b b Chapter 1: Essentials of Geometry distance formula: d Chapter 2: Logic ( x 2 x1 ) 2 ( y 2 y1 ) 2 conditional: If p, then q. pq (Original) qp (Flip) If the car is running, then the key is in the ignition. ( x x 2 ) ( y1 y 2 ) M 1 , 2 2 y y1 slope formula: m 2 x x 1 2 converse: If q, then p. collinear: on the same line bisect: cut in half complementary: add up to 90˚ supplementary: add up to 180˚ If Jenelle gets a job, then she can afford a car. If Jenelle can afford a car, then she will drive to school. CAN CONCLUDE: If Jenelle gets a job, she will drive to school. midpoint formula: If the key is in the ignition, then the car is running. inverse: If not p, then not q. ~p~q (Negate) If the car is not running, then the key is not in the car. contrapositive: If not q, then not p. ~q~p (Flip & Negate) If the key is not in the car, then the car is not running. Law of Syllogism: If a, then b. If b, then c. If a, then c. Law of Detachment: If the hypothesis is true, then the conclusion is true. If the measure of an angle is greater than 90˚ the angle is obtuse. The m<A is 120˚. CAN CONCLUDE: <A is obtuse. Chapter 3: Parallel and Perpendicular Lines <1 & <4 are vertical angles <1 & <5 are corresponding angles <1 & < 8 are alternate exterior angles <3 &<6 are alternate interior angles <3 & < 5 are consecutive interior angles =180 Chapter 4: Congruent Triangles Triangle Interior Angle Sum: 180° Exterior Angle Theorem: m<1=m<2+m<3 Proving Triangles Congruent: Side-Side-Side Side-Angle-Side Angle-Side-Angle Angle-Angle-Side parallel lines: lines that do not intersect and are coplanar (same slopes) perpendicular lines: lines that form a 90˚ angle (slopes are opposite reciprocals) skew lines: lines that do not intersect and are not in the same plane Chapter 5: Relationships Within Triangles Hypotenuse-Leg midsegment- segment parallel to its opposite side and half the length of its opposite side. DE ll AC and DE *** Remember! No swearing or calling AAA Chapter 6: Similar Triangles 1 AC similar: two triangles that have the same angle measures and all corresponding 2 sides have the same ratio (scale factor) perpendicular bisector- a segment, ray, line, or plane that is perpendicular to a segment at its midpoint. Proving Triangles Similar: Perpendicular Bisector Theorem- If a point is ON the Angle-Angle Similarity Side-Side-Side Similarity perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the bisector of AB , then CA=CB. angle bisector- a segment, ray, line, or plane that cuts an angle in half. Angle Bisector Theorem- If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Ratios of ALL of the corresponding sides are equal Side-Angle-Side Similarity Ratios of the corresponding sides are equal, and the included angles are REMEMBER! If two triangles are similar and drawn on top of each other, REDRAW them separately!!! Then set up the proportion. AD bisects <BAC and DB AB and DC AC , then DB=DC. If To form a triangle, the sum of two sides must be greater than the third side. The longest side is opposite the largest angle; smallest angle is opposite the shortest side. If two sides are congruent, the triangle with the biggest angle between the sides has the bigger third side. Don’t forget to get at least 8 hours of sleep the night before the SOL! Also eat a healthy light breakfast so you aren’t focused on your rumbling tummy!