Proving the Pythagorean Theorem Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Proving the Pythagorean Theorem © Boardworks 2012

Information Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Information © Boardworks 2012

Identify the hypotenuse Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Identify the hypotenuse Teacher notes Use this exercise to identify the hypotenuse in right triangles in various orientations. Press the forward arrow to reveal another triangle. There are 12 in total. © Boardworks 2012

The Pythagorean theorem Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem The Pythagorean theorem The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. The area of the largest square is c × c or c2. c2 The areas of the smaller squares are a2 and b2. c a2 a Teacher notes This slide shows how the Pythagorean theorem can be written as a relationship between the side lengths of a triangle with legs a and b and hypotenuse c. Ask students to tell you what a2 is equal to. (c2 – b2). Ask students to tell you what b2 is equal to. (c2 – a2). The area of any similar shapes may by drawn on the sides of a right triangle. The area of the shape drawn on the hypotenuse will be equal to the sum of the areas of the shapes drawn on the two shorter sides. The Pythagorean theorem can be written as: b b2 c2 = a2 + b2 © Boardworks 2012

Showing the Pythagorean theorem Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Showing the Pythagorean theorem Teacher notes Drag the vertices of the triangle to change the lengths of the sides and rotate the right triangle. Ask a volunteer to come to the board and use the pen tool to demonstrate how to find the area of each square. For tilted squares this can be done by using the grid to divide the squares into triangles and squares. Alternatively, a larger square can be drawn around the tilted square and the areas of the four surrounding triangles subtracted. Press the question marks to reveal the areas of the squares and verify that the area of the largest square is always equal to the sum of the areas of the squares on the shorter sides. Red question marks show areas; blue question marks show lengths. This activity can also be used to review the distance formula. Mathematical practices 2) Reason abstractly and quantitatively. Students should be able to make sense of quantities and use them to create a coherent representation of the problem at hand. © Boardworks 2012

Right triangles Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Right triangles Teacher notes Review the definition of a right triangle. Explain that the hypotenuse is the leg that does not touch the right angle. Students might need to be reminded that “adjacent” means “next to,” and this should help them label the legs. Point out that if we labeled the sides with respect to the other acute angle, the opposite and adjacent legs would be reversed. Ask students to explain why no other angle in a right triangle can be larger than or equal to the right angle. Ask students to tell you the sum of the two smaller angles in a right triangle. Recall that the sum of the angles in a triangle is always equal to 180°. Recall also that two angles that add up to 90° are called complementary angles. Conclude that the two smaller angles in a right triangle are complementary angles. Mathematical practices 6) Attend to precision. Students should understand and use appropriate terminology for the legs of a triangle. © Boardworks 2012

A proof of the Pythagorean theorem Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem A proof of the Pythagorean theorem Teacher notes There are many proofs of the Pythagorean theorem. Students could be asked to research these on the internet and make posters to show them. In this proof we can see that the area of the two large squares is the same: (a + b)2 Press the question marks to reveal each length. Both of these squares contain four identical triangles with side lengths a, b and c. In each of the large squares the four triangles are arranged differently to show visually that the area of the square with side length c is equal to the sum of the areas of the squares with side lengths a and b. This can be shown more formally considering the first arrangement. The area of the square with side length c is equal to the area of the large square with side length (a + b), minus the area of the four right triangles. The area of the four right triangles is 4 × ½ab = 2ab. c2 = (a + b)2 – 2ab Expanding, c2 = a2 + 2ab + b2 – 2ab c2 = a2 + b2 Mathematical practices 2) Reason abstractly and quantitatively. Students should be able to make sense of quantities and use them to create a coherent representation of the problem at hand. © Boardworks 2012

An altitude of a triangle Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem An altitude of a triangle What is an altitude of a triangle? Z An altitude of a triangle is a perpendicular line segment from a side to the opposite vertex. W All triangles have three altitudes. This figure shows one altitude of a right triangle. Y X The Pythagorean theorem can be proved using altitudes and similar triangles. © Boardworks 2012 8

Proof using similarity (1) Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Proof using similarity (1) Show that the three triangles in this figure are similar. The altitude to the hypotenuse of a right triangle creates three right triangles: △XYZ, △WXY and △WYZ. ∠XYZ and ∠XWY are both right angles by definition: Z by the reflexive property: ∠ZXY ≅ ∠WXY by the AA similarity postulate:  △XYZ ~ △WXY W ∠XYZ and ∠YWZ are both right angles by definition: Teacher notes Some students have difficulty discerning shapes within shapes. Call volunteers to the board to trace out the shape of each of the three triangles. See the Similarity presentation for more information on similar triangles. Mathematical practices 3) Construct viable arguments and critique the reasoning of others. Students should be able to analyze situations by breaking them down and build a logical progression of statements to explore the truth of their conjectures. Y X by the reflexive property: ∠XZY ≅ ∠WZY by the AA similarity postulate:  △XYZ ~ △WYZ by the transitive property of congruence:  △WXY ~ △WYZ © Boardworks 2012 9

Proof using similarity (2) Boardworks High School Geometry (Common Core) Proving the Pythagorean Theorem Proof using similarity (2) Show that a2 + b2 = (c + d)2 using similar triangles. △XYZ ~ △YWZ ~ △XWY Z △XYZ is similar to △XWY: c b d = b2 = d(c+d) W c+d b a d △XYZ is similar to △YWZ: a c = a2 = c(c+d) c+d a Y b X adding the two equations: a2 + b2 = c(c+d) + d(c+d) Teacher notes The triangles are listed at the top such that their corresponding vertices are shown in the same order to help students figure out which segments are similar to which. Mathematical practices 3) Construct viable arguments and critique the reasoning of others. Students should be able to analyze situations by breaking them down and build a logical progression of statements to explore the truth of their conjectures. = c2 + cd + dc + d2 = c2 + 2cd + d2  = (c+d)2 (c + d) is the hypotenuse of △XYZ, which proves the Pythagorean theorem. © Boardworks 2012 10