Appendix C Some Geometric Facts about Triangles and Parallelograms This appendix contains some formulas and results from geometry that are impor- tant in the study of trigonometry. Circles For a circle with radius r: r Circumference: C 2r D Area: A r2 D Triangles The sum of the measures of the three angles of a triangle is 180 . ı A triangle in which each angle has a measure of less that 90 is called an ı acute triangle. A triangle that has an angle whose measure is greater than 90ı is called an obtuse triangle. A triangle that contains an angle whose measure is 90ı is called a right triangle. The side of a right triangle that is oppositethe right angle is called 424 Appendix C. Results from Geometry 425 the hypotenuse, and the other two sides are called the legs. An isosceles triangle is a triangle in which two sides of the triangle have equal length. In this case, the two angles across from the two sides of equal length have equal measure. An equilateral triangle is a triangle in which all three sides have the same length. Each angle of an equilateral triangle has a measure of 60ı. Right Triangles The sum of the measures of the two acute angles of a right triangle is 90 . In the diagram on the right, ı β ˛ ˇ 90ı. C D c a The Pythagorean Theorem. In a right triangle, the square of the hypotenuse is equal to the sum of the α γ squares of the other twosides. In the diagram on the b right, c2 a2 b2. D C Special Right Triangles A right triangle in which both acute angles are 45ı. For this type of right triangle, the lengths of the two legs are equal. So if c is the length of the hypotenuse and x is the length of each of the legs, then by the Pythagorean Theorem, c2 x2 x2. Solving this equation for x, we obtain D C 2x2 c2 D o c2 45 x2 D 2 c x c2 x o D s 2 45 x c p2 x c D p2 D 2 426 Appendix C. Results from Geometry A right triangle with acute angles of 30 and 60 . ı ı We start with an equilateral tri- o 30 o angle with sides of length c. 30 c By drawing an angle bisector c x at one of the vertices, we cre- o ate two congruent right triangles 60 o 60 with acute angles of 30 and 60 . c c ı ı 2 2 c 2 c2 x2 This means that the third side of D C 2 2 Á each of these right triangles will 2 2 c c x c have a length of . If the length D 4 2 3c2 of the altitudeis x, then usingthe x2 Pythagorean Theorem, we obtain D 4 3c2 p3 x c D s 4 D 2 Similar Triangles Two triangles are similar if the three angles of one triangle are equal in measure to the three angles of the other triangle. The following diagram shows similar triangles ABC and DEF . We write ABC DEF . 4 4 4 4 C γ F a γ b e d α β α β DEf A c B The sides of similar triangles do not have to have the same length but they will be proportional. Using the notation in the diagram, this means that a b c : d D e D f Appendix C. Results from Geometry 427 Parallelograms We use some properties of parallelo- grams in the study of vectors in Sec- D C β α tion 3.5.A parallelogram is a quadri- lateral with two pairs of parallel sides. We will use the diagram on the right α β to describe some properties of paral- AB lelograms. Opposite sides are equal in length. In the diagram, this means that AB DC and AD BC: D D As shown in the diagram, opposite angles are equal. That is, DAB BCD and ABC CDA: † D † † D † The sum of two adjacent angles is 180 . In the diagram, this means that ı ˛ ˇ 180 : C D ı.