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Transcript
Unit 5: Similarity By Sara Stanley & Leon Chen ~ Polygons Regular Polygons ● 3 or more segments that intersect exactly two sides; one per endpoint ● Can be convex - No sides contain a point inside a polygon ● Can be concave - Sides contain point(s) inside polygon Diagonal - Segment that joins two non-collinear points Center of polygon Apothem - Distance from center to side of polygon A Regular Polygon Must Be: ● Equilateral - All sides congruent ● Equiangular - All angles congruent ● Cannot be concave because the interior angles will not be the same Connection: Regular polygons are found a lot in Unit 11, where we found area and used them to find volume of prisms. Polygon Formulas N is the number of sides in a polygon; Y is the measure of each exterior angle Sum of the interior angles of a polygon = (N - 2) x 180° The number of diagonals in a polygon = 1/2 x N(N-3) N x Y = 360, 360/Y = N, 360/N = Y Connection: When finding the perimeter of regular polygons, you also use (n) as the variable for number of sides. Find the measure of an exterior angle of a dodecagon (12). 360/12 = 30° Common Mistake: Be aware when they ask for the SUM of the measures of the exterior angles. ALWAYS 360° READ CAREFULLY N(180 - 180(N-2)/N) 180N - (180N - 360) 360° Ratios and Proportions Find the proportion of: • A ratio is a comparison of 2 numbers using division 1 • Ratio can be expressed as A to B, A/B, or A:B • A proportion is an equation that states that 2 ratios are equal For example: 4 8 7 14 Connection: We also use ratios when talking about trigonometry. For example, cosine is the ratio of the lengths of the adjacent side to the hypotenuse. x-4 x+3 4x 4x - 18 = x2 - x -12 18 x2 -5x + 6 = 0 x = 2, 3 -----> Proportion is 1/5, 1/6 Common Mistake: BE SURE TO PLUG IN BOTH X’s Remember to check both answers in a quadratic problem Similar Triangles Triangles that have the same angle measurements and the same ratio of side lengths all the way around 4 45° 5 8 45° 10 65° 70° 65° 14 7 Common Mistake: When trying to identify if two triangles are similar, sometimes two side lengths have the same ratio, but the last one does not. To avoid this, remember the ways to prove similarity: SAS, SSS, AA No Choice Theorem 40° 40° 60° 60° If two angles of one triangle are equal to two angles of another triangle then then the third angle of both triangles must be congruent. Angle Bisector Theorem If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides. Given AC bisects ∠BAD, you can say BC ≌ AB CD AD B C D A Side Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally. C Given BD ∥ AE, you can prove that CB ≌ CD CA CE ! Solve for AE if DE = x, CE = 16, CA = 2x, BA = 8 and BD = 12 B D 16 - x 16 2x - 8 2x 32x - 2x2 = 32x - 128 x2 - 64 = 0 -------> x = 8 Ratio is now ½ ----> AE is 24 A E Real Life Usage At the same time, many geometric theorems are used in engineering tasks, one example being this bike. Using the Angle Bisector Theorem, we could find the exact length of metal needed to construct that part of the bike. Similar figures along with ratios and proportions play a vital role in engineering and architecture. Imagine designing a scale model of a bridge. To understand each angle and proportional length of the parts of the bridge is vital to the actual structure. As a fashion/fabric designer, working with shapes and patterns is common. It is important to recognize certain properties of regular polygons. For example, finding the perimeter of a shape, finding the amount of area in a piece of fabric, etc. COMMUNICATION we shall be okay do you want to go first? I do...