Objectives Use properties of 45° - 45° - 90° triangles. Use properties of 30° - 60° - 90° triangles.
Perfect Squares The terms of the following sequence: 1, 4, 9, 16, 25, 36, 49, 64, 81… 1 2,2 2,3 2,4 2, 5 2, 6 2, 7 2, 8 2, 9 2 … Perfect Squares These numbers are called the Perfect Squares.
Square Roots square root The number r is a square root of x if r 2 = x. This is usually written Any positive number has two real square roots, one positive and one negative, √ x and -√ x √ 4 = 2 and -2, since 2 2 = 4 and (-2) 2 = 4 principal square root. The positive square root is considered the principal square root.
Properties of Square Roots Properties of Square Roots ( a, b > 0) Product Property Quotient Property
Simplifying Square Root The properties of square roots allow us to simplify radical expressions. A radical expression is in simplest form when: 1. The radicand has no perfect-square factor other than 1 2. There’s no radical in the denominator
Simplest Radical Form Like the number 3/6, is not in its simplest form. Also, the process of simplification for both numbers involves factors. Method 1: Factoring out a perfect square.
Simplest Radical Form In the second method, pairs of factors come out of the radical as single factors, but single factors stay within the radical. Method 2: Making a factor tree.
Simplest Radical Form This method works because pairs of factors are really perfect squares. So 5·5 is 5 2, the square root of which is 5. Method 2: Making a factor tree.
Practice Express each square root in its simplest form by factoring out a perfect square or by using a factor tree.
Your Turn: Simplify the expression.
Example 1 Evaluate, and then classify the product. 1. ( √5)(√5) = 2. (2 + √5)(2 – √5) =
Conjugates conjugates The radical expressions a + √ b and a – √ b are called conjugates. The product of two conjugates is always a rational number
Example 2 Identify the conjugate of each of the following radical expressions: 1. √ – √11 3. √13 + 9
Rationalizing the Denominator Recall that a radical expression is not in simplest form if it has a radical in the denominator. How could we use conjugates to get rid of any radical in the denominator and why?
Rationalizing the Denominator We can use conjugates to get rid of radicals in the denominator: rationalizing the denominator The process of multiplying the top and bottom of a radical expression by the conjugate of the denominator is called rationalizing the denominator. Fancy One
Example 3 Simplify the expression.
Your Turn: Simplify the expression.
Side Lengths of Special Right ∆s Right triangles whose angle measures are 45° - 45° - 90° or 30° - 60° - 90° are called special right triangles. The theorems that describe the relationships between the side lengths of each of these special right triangles are as follows:
Investigation 1 This triangle is also referred to as a right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles. In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.
Investigation 1 Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.
Investigation 1 Did you notice something interesting about the relationship between the length of the hypotenuse and the length of the legs in each problem of this investigation?
45 ˚ -45 ˚ -90 ˚ Triangle Theorem 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the legs ℓ are congruent and the length of the hypotenuse ℎ is times the length of a leg. Example: ℓ ℓ ℓ
Procedure: Finding the hypotenuse in a 45°-45°-90° Triangle FFind the value of x BBy the Triangle Sum Theorem, the measure of the third angle is 45 °. The triangle is a 45°-45°- 90° right triangle, so the length x of the hypotenuse is √2 times the length of a leg. 3 3 x 45 °
Procedure: Finding the hypotenuse in a 45°-45°-90° Triangle Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2 3 3 x 45 ° 45°-45°-90° Triangle Theorem Substitute values Simplify
Procedure: Finding a leg in a 45°-45°-90° Triangle Find the value of x. Because the triangle is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°- 90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg. 5 x x
Procedure: Finding a leg in a 45°-45°-90° Triangle 5 x x Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem 5 √2 √2x √2 = 5 x= 5 x= 5√2 2 x= Substitute values Divide each side by √2 Simplify Multiply numerator and denominator by √2 Simplify
Example 4a A. Find x. The given angles of this triangle are 45° and 90°. This makes the third angle 45°, since 180 – 45 – 90 = 45. Thus, the triangle is a 45°-45°-90° triangle.
Example 4a Substitution 45°-45°-90° Triangle Theorem
Example 4b B. Find x. The legs of this right triangle have the same measure, x, so it is a 45°-45°-90° triangle. Use the 45°-45°-90° Triangle Theorem.
Example 4b Substitution 45°-45°-90° Triangle Theorem x = 12 Answer: x = 12
Your Turn: A. Find x. A.3.5 B.7 C. D.
Your Turn: B. Find x. A. B. C.16 D.32
Example 5 Find a. The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle. Substitution 45°-45°-90° Triangle Theorem
Example 5 Multiply. Divide. Rationalize the denominator. Divide each side by
Your Turn: Find b. A. B.3 C. D.
Cartoon Time
Investigation 2 The second special right triangle is the 30 ˚ -60 ˚ -90 ˚ right triangle, which is half of an equilateral triangle. Let’s start by using a little deductive reasoning to reveal a useful relationship in 30 ˚ -60 ˚ -90 ˚ right triangles.
Investigation 2 Triangle ABC is equilateral, and segment CD is an altitude. 1. What are m ∠ A and m ∠ B? 2. What are m ∠ ADC and m ∠ BDC? 3. What are m ∠ ACD and m ∠ BCD? 4. Is Δ ADC ≅ Δ BDC? Why? 5. Is AD=BD? Why?
Investigation 2 Notice that altitude CD divides the equilateral triangle into two right triangles with acute angles that measure 30° and 60°. Look at just one of the 30 ˚ - 60 ˚ -90 ˚ right triangles. How do AC and AD compare?Conjecture: In a 30°-60°-90° right triangle, if the side opposite the 30° angle has length s, then the hypotenuse has length -?-. 2s
Investigation 2 Find the length of the indicated side in each right triangle by using the conjecture you just made
Investigation 2 Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side. 6 6√36√3 8 4√34√3 10 5√35√ √ 3 4
Investigation 2 You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture. In a 30 ˚ -60 ˚ -90 ˚ triangle: short side = s hypotenuse = 2s long side = s √ 3
30 ˚ -60 ˚ -90 ˚ Triangle Theorem 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the length of the hypotenuse ℎ is 2 times the length of the shorter leg s, and the length of the longer leg ℓ is √ 3 times the length of the shorter leg. Example: s 2s s√3s√3
Procedure: Finding side lengths in a 30°- 60°-90° Triangle Find the values of s and t. Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg. 30 ° 60 °
Procedure: Side lengths in a 30°-60°-90° Triangle Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem 5 √3 √3s √3 = 5 s= 5 s= 5√3 3 s= Substitute values Divide each side by √3 Simplify Multiply numerator and denominator by √3 Simplify 30 ° 60 °
The length t of the hypotenuse is twice the length s of the shorter leg. Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem t2 ∙ 5√3 3 = Substitute values Simplify 30 ° 60 ° t 10√3 3 =
Example 6 Find x and y. The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle.
Example 6 Find the length of the longer side. Substitution Simplify. 30°-60°-90° Triangle Theorem
Example 6 Find the length of hypotenuse. Substitution Simplify. 30°-60°-90° Triangle Theorem Answer: x = 4,
Your Turn: Find BC. A.4 in. B.8 in. C. D.12 in.
Your Turn: Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends. A. B.10 C.5 D.
Two Special Right Triangles ℓ√2ℓ√2 ℓ ℓ 2s s√3s√3 s
45°-45°-90° Special Right Triangle In a triangle 45°-45°-90°, the hypotenuse is times as long as a leg. 45° Hypotenuse ℓ ℓ ℓ Leg Example: 45° 5 cm 5 cm
30°-60°-90° Special Right Triangle In a triangle 30°-60°-90°, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. 30° 60° Hypotenuse s 2s s Longer Leg Shorter Leg Example: 30° 60° 10 cm 5 cm
Find the value of a and b. 60° 7 cm a b Step 1: Find the missing angle measure.30° Step 2: Decide which special right triangle applies. 30°-60°-90° Step 3: Match the 30°-60°-90° pattern with the problem. 30° 60° s 2s a = cm b = 14 cm Step 5: Solve for a and b Step 4: From the pattern, we know that s = 7, b = 2s, and a = s.
Find the value of a and b. 45° 7 cm a b Step 1: Find the missing angle measure.45° Step 2: Decide which special right triangle applies. 45°-45°-90° Step 3: Match the 45°-45°-90° pattern with the problem. 45° ℓ ℓ ℓ Step 4: From the pattern, we know that ℓ = 7, a = ℓ, and b = ℓ. a = 7 cm b = 7 cm Step 5: Solve for a and b