Warm Up: Find the geometric mean of: a) 12 and 18b) 54 and 36 c) 25 and 49

• Warm Up:

Find the geometric mean of:

a) 12 and 18 b) 54 and 36 c) 25 and 49

• TSWBAT use the lengths of 2 sides of a right triangle in order to find the length of the 3rd side.

• HW: pg 436-439 #1-9 odd, 25, 31, 33, 49

Today’s Agenda• Warm Up

• Anticipatory Set

• Discuss Pythagorean Theorem

• Guided Practice of Pythagorean Theorem

• Discuss the converse of the Pythagorean Theorem

• Guided Practice of The Converse of the Pythagorean Theorem

• Do practice problems

• Closure

Anticipatory Set

• You are at one corner of a football field, and your friend is at the opposite corner. Is it shorter to walk diagonally across the field or to walk around it?

Today’s Agenda

• Math Review

• Anticipatory Set

• Discuss Pythagorean Theorem

• Guided Practice of Pythagorean Theorem

• Discuss the converse of the Pythagorean Theorem

• Guided Practice of The Converse of the Pythagorean Theorem

• Do practice problems

• Closure

Pythagorean Theorem

• If a triangle is a right triangle, then the square of the hypotenuse is equal to the sum of the squares of the lengths of both legs.

• If a triangle is a right triangle, then c2 = a2 + b2.

EXAMPLE 1 Find the length of a hypotenuse

SOLUTION

Find the length of the hypotenuse of the right triangle.

(hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem

x2 = 62 + 82

x2 = 36 + 64

x2 = 100

x = 10 Find the positive square root.

Substitute.

Multiply.

Add.

GUIDED PRACTICE for Example 1

Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form.

1.

GUIDED PRACTICE for Example 1

Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form.

2.

EXAMPLE 2 Standardized Test Practice

GUIDED PRACTICE for Example 2

The top of a ladder rests against a wall, 23 feet above the ground. The base of the ladder is 6 feet away from the wall. What is the length of the ladder?

3.

Math Review Day 1

• TSWBAT use the lengths of 2 sides of a right triangle in order to find the length of the 3rd side.

• HW: Pg 436-439 #12-23 Odd, 27, 28, 35

GUIDED PRACTICE for Example 2

The Pythagorean Theorem is only true for what type of triangle?4.

Pythagorean Triples:

3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

EXAMPLE 3 Find the area of an isosceles triangle

SOLUTION

Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.

STEP 1

Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown.

EXAMPLE 3 Find the area of an isosceles triangle

Use the Pythagorean Theorem to find the height of the triangle.

STEP 2

Pythagorean Theorem

Substitute.

Multiply.

Subtract 25 from each side.

Find the positive square root.

c2 = a2 + b2

12 = h

132 = 52 + h2

169 = 25 + h2

144 = h2

EXAMPLE 3 Find the area of an isosceles triangle

Find the area.

STEP 3

= (10) (12) = 60 m21

2

ANSWER

The area of the triangle is 60 square meters.

Area =1

2(base) (height)

GUIDED PRACTICE for Example 3

Find the area of the triangle.5.

GUIDED PRACTICE for Example 3

Find the area of the triangle.6.

ClosureClosure

Write down what changes you will make in class this semester to be successful.

Warm Up

TSWBAT use the lengths of 2 sides of a right triangle in order to find the length of the 3rd side.

HW: Pg 444-447 #1-29 EOO

Today’s AgendaToday’s AgendaMath ReviewMath ReviewAnticipatory SetAnticipatory SetDiscuss Pythagorean TheoremDiscuss Pythagorean TheoremGuided Practice of Pythagorean TheoremGuided Practice of Pythagorean TheoremDiscuss the converse of the Pythagorean Discuss the converse of the Pythagorean

TheoremTheoremGuided Practice of The Converse of the Guided Practice of The Converse of the

Pythagorean TheoremPythagorean TheoremDo practice problems Do practice problems ClosureClosure

Converse of the Pythagorean Converse of the Pythagorean TheoremTheorem

If the square of the hypotenuse is equal to the sum of the squares of the lengths of both legs, then the triangle is a right triangle.

If c2 = a2 + b2 , then the triangle is a right triangle.

If c2 < a2 + b2 , then the triangle is an acute triangle

If c2 > a2 + b2 , then the triangle is an obtuse triangle

EXAMPLE 1 Verify right triangles

Tell whether the given triangle is a right triangle.a. b.

Let c represent the length of the longest side of the triangle. Check to see whether the side lengths satisfy the equation c2 = a2 + b2.

9 34 81 + 225=?

=?(3 34)2 92 + 152a.

306 = 306

b. 262 222 + 142=?

676 484 + 196=?

676 = 680

The triangle is a right triangle.

The triangle is not a right triangle.

GUIDED PRACTICE for Example 1

1.

4,4 3 , 8

Tell whether a triangle with the given side lengths is a right triangle.

2. 10, 11, and 14

3.

5,6 and 61

EXAMPLE 2 Classify triangles

Can segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right, or obtuse?

SOLUTION

STEP 1Use the Triangle Inequality Theorem to check that the segments can make a triangle.

4.3 + 5.2 = 9.59.5 > 6.1

The side lengths 4.3 feet, 5.2 feet, and 6.1 feet can form a triangle.

EXAMPLE 2 Classify triangles

STEP 2Classify the triangle by comparing the square of the length of the longest side with the sum of squares of the lengths of the shorter sides.

c 2 ? a 2 + b2

6.12 ? 4.3 2 + 5.22

37.21 < 45.53

37.212 ? 18.49 2 + 27.042

The side lengths 4.3 feet, 5.2 feet, and 6.1 feet form an acute triangle.

Compare c 2 with a2 + b2.

Substitute.

Simplify.

c 2 is less than a2 + b2.

EXAMPLE 3 Use the Converse of the Pythagorean Theorem

CatamaranYou are part of a crew that is installing the mast on a catamaran. When the mast is fastened properly, it is perpendicular to the trampoline deck. How can you check that the mast is perpendicular using a tape measure? SOLUTION

To show a line is perpendicular to a plane you must show that the line is perpendicular to two lines in the plane.

EXAMPLE 3 Use the Converse of the Pythagorean Theorem

Think of the mast as a line and the deck as a plane. Use a 3-4-5 right triangle and the Converse of the Pythagorean Theorem to show that the mast is perpendicular to different lines on the deck.

First place a mark 3 feet up the mast and a mark on the deck 4 feet from the mast.

Use the tape measure to check that the distance between the two marks is 5 feet. The mast makes a right angle with the line on the deck.

Finally, repeat the procedure to show that the mast is perpendicular to another line on the deck.

GUIDED PRACTICE for Examples 2 and 3

4. Show that segments with lengths 3, 4, and 6 can form a triangle and classify the triangle as acute, right, or obtuse.

GUIDED PRACTICE for Examples 2 and 3

5. WHAT IF? In Example 3, could you use triangles with side lengths 2, 3,and 4 to verify that you

have perpendicular lines?

Today’s AgendaToday’s AgendaMath ReviewMath ReviewAnticipatory SetAnticipatory SetDiscuss Pythagorean TheoremDiscuss Pythagorean TheoremGuided Practice of Pythagorean TheoremGuided Practice of Pythagorean TheoremDiscuss the converse of the Pythagorean Discuss the converse of the Pythagorean

TheoremTheoremGuided Practice of The Converse of the Guided Practice of The Converse of the

Pythagorean TheoremPythagorean TheoremDo practice problems Do practice problems ClosureClosure

Workbook Workbook pg 124-129 every 5th problempg 124-129 every 5th problem