Power Point for 1/24

Bell Work 1/24 For Exercises 1 and 2, find the value of x. Give your answer in simplest radical form. 1. 2. Simplify each expression. 3. 4.

Pythagorean Triple A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

Identifying Pythagorean Triples 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. Find the positive square root. 50 = c The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they form a Pythagorean triple.

Converse of Pythagorean Theorem The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

Pythagorean Inequalities Theorem You can also use side lengths to classify a triangle as acute or obtuse. A B C c b a

Pythagorean Inequality Thm. To understand why the Pythagorean inequalities are true, consider ∆ABC.

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

Classifying Triangles Example 5 Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 7, 10 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 5, 7, and 10 can be the side lengths of a triangle.

Example 5 continued Step 2 Classify the triangle. c2 = a2 + b2 ? Compare c2 to a2 + b2. 102 = 52 + 72 ? Substitute the longest side for c. 100 = 25 + 49 ? Multiply. 100 > 74 Add and compare. Since c2 > a2 + b2, the triangle is obtuse.

Example 6: Classifying Triangles Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 8, 17 Step 1 Determine if the measures form a triangle. Since 5 + 8 = 13 and 13 > 17, these cannot be the side lengths of a triangle.

Example 7: Classifying Triangles Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7, 12, 16 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 7, 12, and 16 can be the side lengths of a triangle.

Example 7 continued Step 2 Classify the triangle. c2 = a2 + b2 ? Compare c2 to a2 + b2. 162 = 122 + 72 ? Substitute the longest side for c. 256 = 144 + 49 ? Multiply. 256 > 193 Add and compare. Since c2 > a2 + b2, the triangle is obtuse.

Applying Special Right Triangles Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°- 60°- 90° triangles.

Special Right Triangles A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle. A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° triangle.

45-45-90 Theorem

Example using 45-45-90 Thm. Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 8.

Example 1 continued Rationalize the denominator.

Example 2 Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of x = 20 Simplify.

Example 3 Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 16. Rationalize the denominator.

Story Problem Jana is cutting a square of material for a tablecloth. The table’s diagonal is 36 inches. She wants the diagonal of the tablecloth to be an extra 10 inches so it will hang over the edges of the table. What size square should Jana cut to make the tablecloth? Round to the nearest inch. Jana needs a 45°-45°-90° triangle with a hypotenuse of 36 + 10 = 46 inches.

Applying Special Right Triangles A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.

Example 4 Find the values of x and y. Give your answers in simplest radical form. 22 = 2x Hypotenuse = 2(shorter leg) 11 = x Divide both sides by 2. Substitute 11 for x.

Example 5 Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. Hypotenuse = 2(shorter leg). y = 2x Simplify.

Example 6 Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) Divide both sides by 2. Substitute for x. y = 27

Example 7 Find the values of x and y. Give your answers in simplest radical form. y = 2(5) Simplify. y = 10

Example 8 Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) 24 = 2x Divide both sides by 2. 12 = x Substitute 12 for x.

Example 9: Story Problem An ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long? Step 1 The equilateral triangle is divided into two 30°-60°-90° triangles. The height of the triangle is the length of the longer leg.

Example 9 continued Step 2 Find the length x of the shorter leg. Hypotenuse = 2(shorter leg) 6 = 2x 3 = x Divide both sides by 2. Step 3 Find the length h of the longer leg. The pin is approximately 5.2 centimeters high. So the fastener will fit.

Example 10 What if…? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth. Step 1 The equilateral triangle is divided into two 30º-60º-90º triangles. The height of the triangle is the length of the longer leg.

Ex 10 continued Step 2 Find the length x of the shorter leg. Rationalize the denominator. Step 3 Find the length y of the longer leg. Hypotenuse = 2(shorter leg) y = 2x Simplify. Each side is approximately 34.6 cm.