Good Morning!! You will have the 1st 20 minutes of class to work on ACC math quietly! We’ll then be taking notes. The goal is to get 7.1, 7.2 and 7.3 done today, we can do it!! As you work on ACC math I will be passing back your tests 

7.1 Areas of Parallelograms and Triangles

SWBAT… To find the area of a parallelogram To find the are of a triangle

Area Formulas Parallelogram A = bh Triangle A = ½ bh

Example 1 Find the area of the parallelogram 13 12 10

Example 2 Find the area of the quadrilateral ABCD if AC = 35, BF = 18 and DE = 10 B E F C A D

7.2 The Pythagorean Theorem and its Converse

SWBAT… To use the Pythagorean theorem To use the converse of the Pythagorean Theorem

Reminder… In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2 a b c

Real-world connection… A baseball diamond is a square with 90-ft sides. Home plate and second base are at opposite vertices of the square. About how far from home plate is second base?

Pythagorean Triples A set of nonzero whole numbers a, b and c that satisfy the equation a2 + b2 = c2 Common triples to remember… 3,4,5 5,12,13 8,15,17 7,24,25

Converse of the Pythagorean Theorem… The sum of the squares of the measures of two sides of the triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

Right triangle? Determine if the measures of these sides are the sides of a right triangle 40, 41, 48

How to determine if a triangle is right, acute or obtuse?? If c is the longest side and a2 + b2 < c2 the triangle is obtuse a2 + b2 > c2 the triangle is acute a2 + b2 = c2 the triangle is right

Classify the triangles as right, acute or obtuse… 4, 5, 6 15, 8, 21 30, 40, 50

7.3 Special Right Triangles Geometry

SWBAT… Find the side lengths of special right triangles. Use special right triangles to solve real-life problems, such as finding the side lengths of the triangles.

Side lengths of Special Right Triangles Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow.

45°-45°-90° Triangle Theorem √2x 45° In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg. Hypotenuse = √2 ∙ leg

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle Find the value of x By 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 45° x

Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle 3 3 45° x 45°-45°-90° Triangle Theorem Substitute values Simplify Hypotenuse = √2 ∙ leg x = √2 ∙ 3 x = 3√2

Ex. 2: 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

Ex. 2: Finding a leg in a 45°-45°-90° Triangle Statement: Hypotenuse = √2 ∙ leg 5 = √2 ∙ x Reasons: 45°-45°-90° Triangle Theorem Substitute values 5 √2 √2x = Divide each side by √2 5 √2 x = Simplify Multiply numerator and denominator by √2 5 √2 x = 5√2 2 x = Simplify

30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg. 60° 30° √3x Hypotenuse = 2 ∙ shorter leg Longer leg = √3 ∙ shorter leg

Ex. 3: 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°

Ex. 3: Side lengths in a 30°-60°-90° Triangle Statement: Longer leg = √3 ∙ shorter leg 5 = √3 ∙ s Reasons: 30°-60°-90° Triangle Theorem Substitute values 5 √3 √3s = Divide each side by √3 5 √3 s = Simplify Multiply numerator and denominator by √3 5 √3 s = 5√3 3 s = Simplify

Statement: Reasons: Substitute values Simplify The length t of the hypotenuse is twice the length s of the shorter leg. 60° 30° Statement: Hypotenuse = 2 ∙ shorter leg Reasons: 30°-60°-90° Triangle Theorem t 2 ∙ 5√3 3 = Substitute values t 10√3 3 = Simplify

Using Special Right Triangles in Real Life Example 4: Finding the height of a ramp. Tipping platform. A tipping platform is a ramp used to unload trucks. How high is the end of an 80 foot ramp when it is tipped by a 30° angle? By a 45° angle?

Solution: When the angle of elevation is 30°, the height of the ramp is the length of the shorter leg of a 30°-60°-90° triangle. The length of the hypotenuse is 80 feet. 80 = 2h 30°-60°-90° Triangle Theorem 40 = h Divide each side by 2. When the angle of elevation is 30°, the ramp height is about 40 feet.

Solution: When the angle of elevation is 45°, the height of the ramp is the length of a leg of a 45°-45°-90° triangle. The length of the hypotenuse is 80 feet. 80 = √2 ∙ h 45°-45°-90° Triangle Theorem 80 = h Divide each side by √2 √2 56.6 ≈ h Use a calculator to approximate When the angle of elevation is 45°, the ramp height is about 56 feet 7 inches.

Ex. 5: Finding the area of a sign Road sign. The road sign is shaped like an equilateral triangle. Estimate the area of the sign by finding the area of the equilateral triangle. 18 in. h 36 in.

Ex. 5: Solution 18 in. First, find the height h of the triangle by dividing it into two 30°-60°-90° triangles. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 18 inches. h = √3 ∙ 18 = 18√3 30°-60°-90° Triangle Theorem h 36 in. Use h = 18√3 to find the area of the equilateral triangle.

Ex. 5: Solution Area = ½ bh = ½ (36)(18√3) ≈ 561.18 18 in. Area = ½ bh = ½ (36)(18√3) ≈ 561.18 h 36 in. The area of the sign is a bout 561 square inches.

Reminders: Quiz Monday 11/16 Project due Monday 11/16 @ beginning of class ACC math due Monday 11/16 before 3:30 Chapter 7 test Monday 11/23 Tuesday 11/24 second six weeks test Dec 18th practice EOC

Class work… worksheet