Geometry Trigonometry of Right Triangles 2014-06-05 www.njctl.org - PDF document

Slide 1 / 240 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org Slide 2 / 240 Geometry Trigonometry of Right Triangles 2014-06-05 www.njctl.org Slide 3 / 240 Table of Contents Pythagorean Theorem Similarity in Right Triangles Click on a Topic to Special Right Triangles go to that section Trigonometric Ratios Solving Right Triangles Angles of Elevation and Depression Law of Sines and Law of Cosines Area of an Oblique Triangle

Slide 4 / 240 Pythagorean Theorem Return to the Table of Contents Slide 5 / 240 Before learning about similar right triangles and trigonometry, we need to review the Pythagorean Theorem and the Pythagorean Theorem Converse. Slide 6 / 240 Recall that a right triangle is a triangle with a right angle. hypotenuse leg leg The sides form that right angle are the legs. The side opposite the right angle is the hypotenuse. The hypotenuse is also the longest side.

Slide 7 / 240 Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. leg 2 + leg 2 = hypotenuse 2 c a or a 2 + b 2 = c 2 b Slide 8 / 240 Example: Find the length of the missing side of the right triangle. Answer x 9 12 Is the missing side a leg or the hypotenuse of the right triangle? Slide 9 / 240 Solve for x: x 9 9 2 + 12 2 = x 2 81 + 144 = x 2 225 = x 2 12 15 = x  -15 is a extraneous solution, a distance can not equal a negative number. x = 15

Slide 10 / 240 x Example: Find the length of the missing side of the right triangle. 20 28 Is the missing side a leg or the hypotenuse of the right triangle? Answer Slide 11 / 240 1 The missing side is the ________ of the right triangle. x A leg Answer hypotenuse B 6 9 Slide 12 / 240 2 Find the length of the missing side. x 6 9 Answer

Slide 13 / 240 3 The missing side is the _________ of the right triangle. A leg x Answer 15 hypotenuse B 36 Slide 14 / 240 4 Find the length of the missing side. x 15 Answer 36 Slide 15 / 240 Real World Application The safe distance of the base of the ladder from a wall it leans against should be one-fourth of the length of the ladder. Thus, the bottom of a 28-foot ladder should be 7 feet from the wall. How far up the wall 28 feet will a ladder reach? ? 7 feet

Slide 16 / 240 2 2 2 Solve using a + b = c 28 feet ? Answer 7 feet The ladder will reach feet up the wall safely. Slide 17 / 240 Real World Application 84 Answer x 50 The dimensions of a high school basketball court are 84' long and 50' wide. What is the length from one corner of the court to the opposite corner? Slide 18 / 240 5 A NBA court is 50 feet wide and the length from one corner of the court to the opposite corner is 106.5 feet. How long is the court? (Round the answer to the nearest whole number) A 94.03 feet Answer 117.7 feet B 118 feet C 94 feet D

Slide 19 / 240 Pythagorean Theorem Applications The Pythagorean Theorem can also be used in figures that contain right angles. Slide 20 / 240 Example Find the perimeter of the square. 18 cm P sq = 4s note: Before finding the perimeter of the square, we need to first find the length of each side. Slide 21 / 240 Remember, in a square all sides are congruent. Start here: 2 2 2 x + x = 18 18 cm x Answer

Slide 22 / 240 Example 1 Find the area of the triangle. A = bh 2 The base of the triangle is given, but we need to find the height of the triangle. 13 feet 13 feet 10 feet Slide 23 / 240 By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base. 13 feet 13 feet Answer h 5 feet 5 feet Slide 24 / 240 Try this... Find the perimeter of the rectangle. 8 in P rect = 2 l + 2 w Answer 10 in

Slide 25 / 240 6 Find the area of the rectangle. A 120 square feet B 84 square feet Answer t 8 feet e e f 46 square inches 7 C 1 46 square feet D Slide 26 / 240 7 Find the perimeter of the square. (Round to the nearest tenth) A 12.8 cm 25.5 cm B 9 c Answer m 25.6 cm C 36 cm D Slide 27 / 240 8 Find the area of the triangle. Answer 7 inches 7 inches 10 inches

Slide 28 / 240 9 Find the area of the triangle. 7 inches 7 inches Answer 4 inches Slide 29 / 240 Converse of the Pythagorean Theorem If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. B c If c 2 = a 2 + b 2 , then a ABC is a right triangle. A C b Slide 30 / 240 Example Tell whether the triangle is a right triangle . 24 E D Answer 7 Remember c is the 25 longest side F

Slide 31 / 240 Theorem If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. B c a If c 2 > a 2 + b 2 , then ABC is obtuse. A C b Slide 32 / 240 Theorem If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. B c a If c 2 < a 2 + b 2 , then A ABC is acute. C b Slide 33 / 240 Example Classify the triangle as acute, right, or obtuse. 15 13 Answer 17

Slide 34 / 240 10 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 12 B right Answer obtuse C 15 not a triangle D 11 Slide 35 / 240 11 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 5 right B Answer 3 obtuse C not a triangle D 6 Slide 36 / 240 12 Classify the triangle as acute, right, obtuse, or not a triangle. A acute 25 right B Answer obtuse C 20 not a triangle D 19

Slide 37 / 240 13 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. A acute right B Answer obtuse C Slide 38 / 240 14 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. A acute triangle Answer B right triangle obtuse triangle C Slide 39 / 240 Review If c 2 = a 2 + b 2 , then triangle is right. If c 2 > a 2 + b 2 , then triangle is obtuse. If c 2 < a 2 + b 2 , then triangle is acute.

Slide 40 / 240 Similarity in Right Triangles Return to the Table of Contents Slide 41 / 240 There are many proofs to the Pythagorean Theorem. How many do you know? Triangle similarity can be used to prove the Pythagorean Theorem. How? Slide 42 / 240 Theorem The altitude of a right triangle divides the triangle into two smaller triangles that are similar to the original triangle and each other. C A B D CD is the altitude of ABC ABC~ ACD~ CBD

Slide 43 / 240 To prove this, click for Lab 1 - Similar Right Triangles Teacher Notes Therefore, the altitude of a right triangle divides the triangle into two smaller triangles that are similar to the click original triangle and similar to each other. click Slide 44 / 240 C Let's prove the Theorem. The altitude of a right triangle divides the triangle into A B D two smaller triangles that are similar to the original triangle Statements Reasons and each other . Given ABC is a right triangle click Given: ABC is a right triangle is a right angle Given click is the altitude of ABC Def of Altitude click Def of Perp Lines. 2 lines that form a Prove: ABC~ ACD~ CBD is a right angle click rt angle All rt angles are click Reflexive Prop of click ABC ~ ACD click AA~ is a right angle click Def of Perp Lines All rt angles are click Reflexive Prop of click ABC ~ CBD AA~ click ABC~ ACD~ CBD Transitive Prop of ~ click Slide 45 / 240 C Let's sketch the 3 triangle's separately, with the same orientation. A B D B B Match up the angles. C A C C D A D Helpful tip: If you set , then you can assign all the angles a value and easily find the matches. 30 B B C 30 30 30 60 60 60 60 30 60 C D A D C A