RIGHT TRIANGLES AND THE PYTHAGOREAN THEOREM Lilly Weston Curriculum 2085 University of Louisiana at Monroe

Table of Contents What’s a right triangle and what makes it special? A right triangle versus an isosceles What does the Pythagorean Theorem mean? What are a, b, and c? When would you use this theorem? Problems –Is this a right triangle?Is this a right triangle? –Missing side’s length?Missing side’s length? –Word problemWord problem –Will these measurements create a right triangle?Will these measurements create a right triangle? Who created this theorem? Review part one Review part two

What’s a right triangle and what makes it special? Things all triangles share: Three sides Three angles All the angles, when added together, equal 180 ° Only right triangles have: A 90° angle A hypotenuse The three sides are given specific names –a, b, and c The Pythagorean theorem can ONLY be used on these – a 2 + b 2 = c 2

A right triangle versus an isosceles Right triangle Another triangle (isosceles)

What does the Pythagorean theorem mean? The sum of the two areas of the two squares (a and b) equals the area of the square of the hypotenuse (c). –a 2 + b 2 = c 2 Visual proof More info about this is at Khan AcademyKhan Academy

What are a, b, and c? a b c 90 ° Legs – a and b create the 90 ° angle Hypotenuse – c this side will always be on the opposite side of the 90° angle the longest of the three sides

When would you use this theorem? You would use this if you’re unsure if a triangle is a right, if you have two sides of the right triangle and you’re trying to find the length of the third side, or for certain word problems. Example: If a = 5 and b = 9, what will c equal? Example: 7 12 b = ?

What are the steps? Find c if a = 12 and b = 14. Round to the nearest tenth. Step 1: Write the equation down. –a 2 + b 2 = c 2 Step 2: Plug in the information to the correct variable. –(12) 2 + (14) 2 = c 2 Step 3: Solve – this isn’t always the same for each problem.* – = c 2 –340 = c 2 –√(340) = √(c 2) –18.4 = c *For this problem, we had to use the square root on both sides of the equation. Before you put down your answer, make sure to look at how it should be rounded!

Is this a right triangle? Is this a right triangle? a 2 + b 2 = c 2 (8) 2 + (6) 2 = (10) = = 100 YES* *For this problem, we didn’t have to use the square root. All we needed to do was square everything and add on both sides of the equal sign.

What is the length of the missing side? Round to the nearest tenth. 1. a 2 + b 2 = c 2 2. (7) 2 + (3) 2 = c = c 2 54 = c 2 √(54) = √(c 2 ) 7.6 = c 7 3 c

Word Problem 1. A 30 foot ladder is leaning against a wall. The bottom of the ladder is 6 feet from the base of the wall. How tall is the wall? 1.a 2 + b 2 = c 2 2.a 2 + (6) 2 = (30) 2 3.a = a 2 = 864 √(a 2 ) = √(864) a = 29.4 feet 30 feet 6 feet a=?

Will these measurements create a right triangle? a = 6.4, b = 12, c = a 2 + b 2 = c 2 2.(6.4) 2 + (12) 2 = (12.2) = ≠ NO

Who created this theorem? Pythagoras ( B.C.) From the Greek island Samos Died in Metapontum, Italy Because there is little written work of his, some are unsure that he contributed to mathematics at all. Influenced Plato and all of Western Philosophy

Review – part one 1.T or F? You can use this equation on any type of triangle? FALSE; it can only be used on right triangles 2. T or F? Pythagoras was born in Italy and died on a Greek island. FALSE; he was born on a Greek island (Samos) and died in Italy (Metapontum) 3.Could these create a right triangle? a = 12.5, b = 14, c = 11. Do not work out the problem. NO; the hypotenuse, c, must be the longest of the three sides 4. What is the length of the missing side(to the nearest tenth)? 28 a 18 a 2 + b 2 = c 2 a 2 + (18) 2 = (28) 2 a = a 2 = 460 √(a 2 ) = √(460) a = 21.4

Review – part two 1.Where, on this triangle, does each letter belong? 2.What are the difference(s) between a right triangle and an equilateral triangle? 1.R: one 90 ° angle + 2 other angles, side lengths are not necessarily the same length 2.E: three 60° angles, all three sides are the same length 3.What is true of the hypotenuse, c, no matter what the situation? 1.It is always the longest side. c a b