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Transcript
Triangle Properties Phil Plante, Olney, MD 1. 3 sided, closed polygon 2. Angles: a. Sum of interior angles: mA+ mB+ mC = 180o b. Sum of exterior angles = 2*sum of interior angles = 360o c. An exterior angle = the sum of the nonadjacent interior angles 3. Types of triangles: a. Right = one right angle b. Acute = all acute (<90o) angles c. Obtuse = one obtuse angles (>90o) d. Equiangular = Equilateral = 3 equal angles and sides e. Isosceles = 2 equal sides; equal base angles f. Scalene = no equal sides or angles 4. Perimeter and Area: a. Perimeter = a + b + c b. Area = (1/2)(base)(height) = (1/2)bh, where h is an altitude 5. Median: a. Line segment from a vertex to the midpoint of opposite side b. Point of concurrency = Centroid always inside the triangle balance point divides each median into two parts in 2:1 ratio, longer side near the vertex c. Medians divide triangle into multiple triangles that all have the same area 6. Angle Bisector: a. Line from each vertex that bisects the angle b. Point of Concurrent = Incenter always inside equidistant from all sides center of inscribed circle c. Theorem: An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. 7. Perpendicular Bisector: a. Line that is perpendicular to and bisects a side b. Point of concurrency = Circumcenter Outside if obtuse, inside otherwise equidistant from all vertices center of circumcircle through all vertices c. If it hits a vertex, it is also an angle bisector and the triangle is isosceles Triangles 8. Altitudes: a. Segment from a vertex perpendicular to opposite side b. Point of concurrency = Orthocenter. Outside if obtuse; inside otherwise 9. Midline (Triangle Midpoint Theorem): a. Line segment joining the midpoint of one side to the midpoint of another b. The midline is parallel to the third side and one-half its length c. The three midlines form 4 congruent triangles inside the larger one, each similar to the larger triangle with scale factor (k) = ½. 10. Triangle Congruence: same shape, same a. SSS = all three corresponding sides are congruent b. SAS = two sides and the angle between them are congruent c. ASA = two angles and the side between them are congruent d. AAS = two angles and a non-included side e. For right triangles: HL = The hypotenuse and one leg are congruent HA = The hypotenuse and one acute angle are congruent LL = both legs are congruent LA = One leg and one acute angle are congruent f. CPCTC = corresponding parts of congruent triangles are congruent 11. Triangle Similarity: same shape, different size a. Corresponding angles are equal; ratio of all corresponding sides = scale factor k b. AA: if two angles are congruent, the triangles are similar c. SSS: ratios of all corresponding sides equal the scale factor (k): 𝒔𝒊𝒅𝒆 𝒂′ 𝒔𝒊𝒅𝒆 𝒃′ 𝒔𝒊𝒅𝒆 𝒄′ = = =𝒌 𝒔𝒊𝒅𝒆 𝒂 𝒔𝒊𝒅𝒆 𝒃 𝒔𝒊𝒅𝒆 𝒄 d. SAS: Ratio of corresponding sides = k, included angles are equal e. Ratio of perimeters = k f. Ratio of areas = k2 g. Ratio of volumes = k3 12. Triangle Inequality Theorem: a. The sum of the lengths of any two sides of triangle > the length of the third side b. If two angles of a triangle are unequal, their opposite sides are unequal and the larger side will be opposite the larger angle c. If two sides of a triangle are unequal, their opposite angles are unequal and the larger angle will be opposite the larger side 13. Right Triangles: a. Pythagorean Theorem: i. a2 + b2 = c2 ii. If c2 > a2+b2, triangle is obtuse; if c2 < a2+b2, triangle is acute b. Pythagorean triple = right triangle with all three sides being whole numbers i. Can scale up any triple by multiplying by another whole number (e.g., (3,4,5)*2 = (6, 8,10)). Scaled up versions = same family. 2|Page Triangles ii. Common ones (not including same family): (3,4,5), (5,12,13), (7,24,25), (8,15,17), (9.40,41), (11,60,61), … infinitely many more 14. Geometric Mean Theorem a. If an altitude is drawn to the hypotenuse of a right triangle, it creates two similar right triangles, each similar to the original triangle and to one another b. ABD ~ BCD ~ ABC c. In similar triangles, the ratios of two corresponding sides are equal. For triangles ABD and BCD then, we have: 𝐴𝐷 ℎ = ℎ 𝐶𝐷 Cross multiplying: AD*CD = h2 √𝑨𝑫 ∗ 𝑪𝑫 = 𝒉 The altitude h is the geometric mean of lines AD and CD d. Scale Factors (k) for triangles 2 and 3 wrt triangle 1: k21 = 𝑨𝑩⁄𝑨𝑪 k31 = 𝑩𝑪⁄𝑨𝑪 15. Law of Sines and Cosines a. Applies to non-right triangles b. Law of Sines: 𝑺𝒊𝒏(𝑨) 𝒂 = 𝑺𝒊𝒏(𝑩) 𝒃 = 𝑺𝒊𝒏(𝑪) 𝒄 c. Law of Cosines: 𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄𝑪𝒐𝒔(𝑨) 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄𝑪𝒐𝒔(𝑩) 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃𝑪𝒐𝒔(𝑪) 3|Page