Pearson Unit 3 Topic 10: Right Triangles and Trigonometry 10-1: The Pythagorean Theorem and Its Converse Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007 www.themathlab.com

TEKS Focus: (9)(B) Apply the relationships in special right triangles 30-60- 90 and 45-45-90 and the Pythagorean theorem, including Pythagorean triples, to solve problems. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (1)(C) Select tools, including real objects, manipulatives paper and pencil, and technology as appropriate, and techniques, including mental math, estimations, and number sense as appropriate, to solve problems. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (2)(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines. (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

History Pythagoras was born in the late 6th century B.C. on the island of Samos. At an early age Pythagoras developed a passion for learning. On one trip to Egypt, Pythagoras studied a group of people known as the “rope-stretchers.” They were the engineers who built the pyramids. Pythagoras Pic: http://en.wikipedia.org/wiki/Pythagoras Pyramid Pic: www.greenwichmeantime.com

History continued The Rope-Stretchers tied a rope in a circle with 12 evenly spaced knots. If shaped into a triangle with sides 3-4-5 a right triangle would emerge and enable them to lay the corners of the foundations for their buildings accurately. Rope Pic and History: http://www.themathlab.com/Algebra/lines%20and%20distances/pythagor.htm

Copy these in your Flip Book! Another Pythagorean Triple not listed above is 9, 40, 41. Therefore, these are also Pythagorean Triples: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 9, 40, 41 6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50 18, 80, 82 9, 12, 15 15, 36, 39 24, 45, 51 21, 72, 75 27, 120, 123 12, 16, 20 20, 48, 52 32, 60, 68 28, 96, 100 36, 160, 164 Copy these in your Flip Book!

Example in your Flip Book: 8 b 2√7 (2 7 ) 2 + 𝑏 2 = 8 2 4∗7+ 𝑏 2 =64 28+ 𝑏 2 =64 𝑏 2 =36 𝑏=6

Example in your Flip Book: (4 3 ) 2 + 𝑏 2 = (6 5 ) 2 16∗3+ 𝑏 2 =36∗5 48+ 𝑏 2 =180 𝑏 2 =132 𝑏= 132 𝑏= 4 33 𝑏=2 33

Example in your Flip Book: Find RS. 𝑦 2 + 6 2 = 8 2 𝑦 2 +36=64 𝑦 2 =28 𝑦= 28 𝑦= 4 7 𝑦=2 7 𝑅𝑆=2 7 +2 7 𝑅𝑆=4 7

Example in your Flip Book: If AC = 10, then BC = 6 because opposite sides of a rectangle are congruent. And DC = EB = x. Therefore, AB = 10 – 6 = 4. 𝑥 2 + 4 2 = 12 2 𝑥 2 +16=144 𝑥 2 =128 𝑥= 128 𝑥= 64 2 𝑥= 8 2

By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length. Remember!

a2 + b2 < c2 OBTUSE a2 + b2 = c2 RIGHT a2 + b2 > c2 ACUTE Here is a helpful way to remember from theorems: c2 > a2 + b2 OBTUSE c2 = a2 + b2 RIGHT c2 < a2 + b2 ACUTE Or you can remember it this way, and Ms. Doss prefers this way: a2 + b2 < c2 OBTUSE a2 + b2 = c2 RIGHT a2 + b2 > c2 ACUTE

Example in your Flip Book: Tell if the measures, 2, 4, and 10, can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Step 1 Determine if the measures form a triangle. Since 2 + 4 = 6 and 6 is not greater than 10, these cannot be the side lengths of a triangle.

Example in your Flip Book: Tell if the measures, 7, 8, and 9, can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 7, 8, and 9 can be the side lengths of a triangle because 7 + 8 = 15 which is greater than 9. Step 2 Classify the triangle. 7 2 + 8 2 ? 9 2 49+64 ? 81 113> 81 This is an acute triangle.

Example in your Flip Book: Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 16, 30, 34 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 16, 30 and 34 can be the side lengths of a triangle because 16 + 30 is greater than 34. Step 2 Classify the triangle. 16 2 + 30 2 ? 34 2 256+900 ? 1156 1156= 1156 This is a right triangle.

Example in your Flip Book: Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 3 2 ,2 5 ,3 6 Step 1 Determine if the measures form a triangle. Approximate the values to determine a, b, c. In order from above, 4.24 ≈ a, 4.47 ≈ b, 7.35 ≈ c. 4.24 + 4.47 > 7.35, so it does form a triangle. Step 2 Classify the triangle—continued on next slide.

Flip book example continued Step 2 Classify the triangle. 3 2 ,2 5 ,3 6 3 2 2 + 2 5 2 ? 3 6 2 9∗2+4∗5 ?9∗6 18+20 ?54 38 < 54 This is an obtuse triangle.

Example in your Flip Book: Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 8 2 ,5 2 ,6 3 Step 1 Determine if the measures form a triangle. Approximate the values to determine a, b, c. In order from above, 11.31 ≈ c, 7.07 ≈ a, 10.39 ≈ b. 7.07 + 10.39 > 11.31, so it does form a triangle. Step 2 Classify the triangle—continued on next slide.

Flip book example continued Step 2 Classify the triangle. 8 2 ,5 2 ,6 3 5 2 2 + 6 3 2 ? 8 2 2 25∗2+36∗3 ?64∗2 50+108 ?128 158 > 128 This is an acute triangle.

Example in your Flip Book: Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 12 ,2 8 , 18 Step 1 Determine if the measures form a triangle. Approximate the values to determine a, b, c. In order from above, 3.46 ≈ a, 5.66 ≈ c, 4.24 ≈ b. 3.46 + 4.24 > 5.66, so it does form a triangle. Step 2 Classify the triangle—continued on next slide.

Flip book example continued Step 2 Classify the triangle. 12 ,2 8 , 18 12 2 + 18 2 ? 2 8 2 12 +18 ? 4∗8 30 < 32 This is an obtuse triangle.

This is not in Flip Book but it is important for you to know! Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a2 + b2 = c2 42 + b2 = 122 b2 = 128 The side lengths do not form a Pythagorean triple because is not a whole number.

The following examples are not in your Flip Book, but are good examples for you to read through.

Example: 1 (not in flip book) Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem 22 + 62 = x2 Substitute 2 for a, 6 for b, and x for c. 40 = x2 Simplify. Find the positive square root. Simplify the radical.

Example: 2 (not in flip book) Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 Pythagorean Theorem (x – 2)2 + 42 = x2 Substitute x – 2 for a, 4 for b, and x for c. x2 – 4x + 4 + 16 = x2 Multiply. –4x + 20 = 0 Combine like terms. 20 = 4x Add 4x to both sides. 5 = x Divide both sides by 4.

Example: 3 (not in flip book) Dog agility courses often contain a seesaw obstacle. How far above the ground is the dog’s paw? Round to nearest inch.

Example: 4 (not in flip book) According to the recommended safety ratio of 4:1, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4x is the distance in feet from the top of the ladder to the base of the wall.

Example: 4 continued a2 + b2 = c2 (4x)2 + x2 = 302 17x2 = 900 Pythagorean Theorem a2 + b2 = c2 Substitute 4x for a, x for b, and 30 for c. (4x)2 + x2 = 302 Multiply and combine like terms. 17x2 = 900 Now multiply that square root answer by 4 to get the height and then round to the nearest inch! Remember that you round once on a problem, and that it at the very end! Height ≈ 29 inches

Example: 5 (not in flip book)

Example: 6 (not in flip book)

Example: 7 (not in flip book) Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a2 + b2 = c2 Pythagorean Theorem 142 + 482 = c2 Substitute 14 for a and 48 for b. 2500 = c2 Multiply and add. 50 = c Find the positive square root. The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they do form a Pythagorean triple.