7-1 and 7-2: Apply the Pythagorean Theorem Geometry Chapter 7 7-1 and 7-2: Apply the Pythagorean Theorem

Simplifying Radicals When taking the square root, it can be simplified in one of two ways, depending on the number in the root: 1.) The number is a perfect square 2.) The number is not a perfect square

Simplifying Radicals When taking the square root, it can be simplified in one of two ways, depending on the number in the root: 1.) The number is a perfect square EX: 𝟑𝟔 𝟔 2.) The number is not a perfect square

Simplifying Radicals When taking the square root, it can be simplified in one of two ways, depending on the number in the root: 1.) The number is a perfect square EX: 𝟑𝟔 𝟔 2.) The number is not a perfect square EX: 𝟑𝟐 𝟐×𝟏𝟔 𝟒 𝟐

Warm-Up Simplify the following radicals. 1.) 100 2.) 64 3.) 25 4.) 144

Warm-Up Simplify the following radicals. 5.) 45 6.) 24 7.) 27 8.) 31

Apply the Pythagorean Theorem Objective: Students will be able to find side lengths in right triangles, as well as identify triangles, using the Pythagorean Theorem. Agenda Right Triangles Pythagorean Theorem Pythagorean Triples Identify Triangles

Right Triangles The sides of a right triangle named as such: The side opposite the right angle is known as the Hypotenuse The other two sides are known as the Legs Hypotenuse Leg

The Pythagorean Theorem Theorem 7.1 – The Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. 𝒄 𝒃 𝒂 𝑐 2 = 𝑎 2 + 𝑏 2

Example 1 Use the Pythagorean Theorem to find the value of x.

Example 1 Use the Pythagorean Theorem to find the value of x. 𝑎=5 𝑏=12 𝑐=𝑥

Example 1 Use the Pythagorean Theorem to find the value of x. 𝑎=5 𝑏=12 𝑐=𝑥 𝑥 2 = 5 2 + 12 2 𝑥 2 =25+144 𝑥 2 =169 𝑥= 169 =13

Example 2 Use the Pythagorean Theorem to find the value of x.

Example 2 Use the Pythagorean Theorem to find the value of x. 𝑎=𝑥 𝑏=9 𝑐=11

Example 2 Use the Pythagorean Theorem to find the value of x. 11 2 = 𝑥 2 + 9 2 121= 𝑥 2 +81 𝑥 2 =40 𝑥= 40 =2 10 𝑎=𝑥 𝑏=9 𝑐=11

Example 3 Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse 𝟓 𝒙 𝟑

Example 3 Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse 5 2 = 𝑥 2 + 3 2 25= 𝑥 2 +9 𝑥 2 =16 𝑥= 16 =𝟒 𝟓 𝒙 𝟑 𝒙=𝟒 Leg

Example 4 Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse 𝒙 𝟔 𝟒

Example 4 Use the Pythagorean Theorem to find the value of x. Identify x as either a leg or hypotenuse 𝑥 2 = 6 2 + 4 2 𝑥 2 =36+16 𝑥 2 =52 𝑥= 52 =𝟐 𝟏𝟑 𝒙 𝟔 𝟒 𝒙=𝟐 𝟏𝟑 Hypotenuse

Example 5 Find the Area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. 𝟏𝟑 𝟏𝟎

Example 5 Identify the height by labeling it on the drawing. 𝟏𝟑 𝟓 𝒃 𝒉 Recall: Area of a Triangle 𝑨= 𝟏 𝟐 𝒃𝒉

Example 5 Use the Pythagorean Theorem to find the value of h. 𝟏𝟑 𝟓 𝒃 𝒉 13 2 = ℎ 2 + 5 2 169=ℎ 2 +25 ℎ 2 =144 ℎ= 144 =12

Example 5 Solve for area 𝟏𝟑 𝟓 𝟏𝟎 𝟏𝟐 𝐴= 1 2 𝑏ℎ 𝐴= 1 2 (10)(12) 𝑨=𝟔𝟎

Example 6 Find the area of the given triangle. 𝟏𝟖 𝟑𝟎

Example 6 Find the area of the given triangle. 𝟑𝟎 18 2 = ℎ 2 + 15 2 324=ℎ 2 +225 ℎ 2 =99 ℎ= 99 =3 11 𝟏𝟖 𝟏𝟓 𝒉

Example 6 Find the area of the given triangle. 𝟑𝟎 𝐴= 1 2 𝑏ℎ 𝟏𝟓 𝐴= 1 2 (30)(3 11 ) 𝑨=𝟒𝟓 𝟏𝟏 𝟏𝟖 𝟏𝟓 𝒉

Example 6 Find the area of the given triangle. 𝟐𝟔 𝟐𝟎

Example 6 Find the area of the given triangle. 𝟐𝟔 𝟏𝟎 26 2 = ℎ 2 + 10 2 676=ℎ 2 +100 ℎ 2 =576 ℎ= 576 =24 𝟐𝟎 𝒉 𝟏𝟎

Example 6 Find the area of the given triangle. 𝟐𝟔 𝟏𝟎 𝐴= 1 2 𝑏ℎ 𝐴= 1 2 (20)(24) 𝑨=𝟐𝟒𝟎 𝟐𝟎 𝟐𝟒 𝟏𝟎

Pythagorean Triples When a right triangle has side lengths that are all whole numbers, we call them a Pythagorean Triple. Examples of Pythagorean Triples: 3 – 4 – 5 5 – 12 – 13 8 – 15 – 17 7 – 24 – 25

Pythagorean Triples When a right triangle has side lengths that are all whole numbers, we call them a Pythagorean Triple. Examples of Pythagorean Triples: 3 – 4 – 5 6 – 8 – 10 9 – 12 – 15 15 – 20 – 25 12 – 16 – 20 5 – 12 – 13 8 – 15 – 17 7 – 24 – 25 10 – 24 – 26 16 – 30 – 34 14 – 48 – 50 15 – 36 – 39 20 – 48 – 52 24 – 45 – 51

Example 7 Find the value of x on both triangles 𝟐𝟓 𝟕 𝒙 𝒙 𝟖 𝟔

Example 7 Find the value of x on both triangles 𝟐𝟓 𝟕 𝒙 𝒙 𝟖 𝟔 𝒙=𝟏𝟎 𝒙=𝟐𝟒

Example 7 Find the value of x on both triangles 𝟏𝟓 𝒙 𝟏𝟐 𝒙 𝟓 𝟒

Example 7 Find the value of x on both triangles 𝟏𝟓 𝒙 𝟏𝟐 𝒙=𝟗 𝒙 𝟓 𝟒

Example 7 Find the value of x on both triangles 𝒙=𝟗 𝑥 2 = 4 2 + 5 2 𝟏𝟓 𝒙 𝟏𝟐 𝒙=𝟗 𝑥 2 = 4 2 + 5 2 𝑥 2 =16+25 𝑥 2 =41 𝒙= 𝟒𝟏 ≈𝟔.𝟒𝟎𝟑 𝒙 𝟓 𝟒

The Pythagorean Theorem Theorem 7.2 – Converse of the Pythagorean Theorem: If the square of the length on the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If 𝑐 2 = 𝑎 2 + 𝑏 2 , Then ∆𝐴𝐵𝐶 is a right triangle 𝒄 𝒃 𝒂 𝑨 𝑩 𝑪

Example 8 Tell whether the given triangle is a right triangle. 𝟗 𝟑 𝟑𝟒 𝟏𝟓 𝟑 𝟑𝟒 𝟗

Example 8 Tell whether the given triangle is a right triangle. 𝟏𝟓 𝟑 𝟑𝟒 𝟗 (3 34 ) 2 = 15 2 + 9 2 9×34=225+81 𝟑𝟎𝟔=𝟑𝟎𝟔

Example 8 Tell whether the given triangle is a right triangle. 𝟏𝟓 𝟑 𝟑𝟒 𝟗 (3 34 ) 2 = 15 2 + 9 2 9×34=225+81 𝟑𝟎𝟔=𝟑𝟎𝟔 Thus, the triangle is a Right Triangle

Example 9 Tell whether the given triangle is a right triangle. 𝟏𝟒 𝟐𝟐 𝟐𝟔 𝟐𝟐 𝟏𝟒

Example 9 Tell whether the given triangle is a right triangle. 26 2 = 22 2 + 14 2 676=484+196 𝟔𝟕𝟔≠𝟔𝟖𝟎 𝟐𝟔 𝟐𝟐 𝟏𝟒 Thus, the triangle is NOT a Right Triangle

The Pythagorean Theorem Theorem 7.3: If the square of the length on the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle. 𝒄 𝒃 𝒂 𝑨 𝑩 𝑪 If 𝑐 2 < 𝑎 2 + 𝑏 2 , Then ∆𝐴𝐵𝐶 is an acute triangle

The Pythagorean Theorem Theorem 7.4: If the square of the length on the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle. 𝒄 𝒃 𝒂 𝑨 𝑩 𝑪 If 𝑐 2 > 𝑎 2 + 𝑏 2 , Then ∆𝐴𝐵𝐶 is an obtuse triangle

Example 10 Determine if a triangle with side lengths 4.3, 5.2, and 6.1 form a right, acute, or obtuse triangle.

Example 10 Determine if a triangle with side lengths 4.3, 5.2, and 6.1 form a right, acute, or obtuse triangle. 𝑎=4.2 𝑏=5.2 𝑐=6.1 6.1 2 = 5.2 2 + 4.3 2 37.21=27.04+18.49 𝟑𝟕.𝟐𝟏<𝟒𝟓.𝟓𝟑

Example 10 Determine if a triangle with side lengths 4.3, 5.2, and 6.1 form a right, acute, or obtuse triangle. 𝑎=4.2 𝑏=5.2 𝑐=6.1 6.1 2 = 5.2 2 + 4.3 2 37.21=27.04+18.49 𝟑𝟕.𝟐𝟏<𝟒𝟓.𝟓𝟑 Thus, the triangle is an Acute Triangle.

Example 10 Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23 Determine if the triangles with the given side lengths form a right, acute, or obtuse triangle. Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23

Example 10 Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23 Determine if the triangles with the given side lengths form a right, acute, or obtuse triangle. Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23 8 2 = (4 3 ) 2 + 4 2 64=(16×3)+16 𝟔𝟒=𝟔𝟒 Thus, the triangle is a Right Triangle.

Example 10 Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23 Determine if the triangles with the given side lengths form a right, acute, or obtuse triangle. Side Lengths: 4, 4 3 , 8 Side Lengths: 11, 20, 23 8 2 = (4 3 ) 2 + 4 2 64=(16×3)+16 𝟔𝟒=𝟔𝟒 23 2 = 20 2 + 11 2 529=400+121 𝟓𝟐𝟗>𝟓𝟐𝟏 Thus, the triangle is a Right Triangle. Thus, the triangle is an Obtuse Triangle.

𝟐𝟔 𝟏𝟎 𝒙 𝒙 𝟏𝟐 𝟏𝟔 𝟖 𝒙 𝟔

𝟑𝟗 𝟏𝟓 𝒙 𝒙 𝟓 𝟏𝟑 𝒙 𝟒𝟎 𝟑𝟎