Becky Afghani, LBUSD Math Curriculum Office, 2004 Right Triangles

Becky Afghani, LBUSD Math Curriculum Office, 2004

Right Triangles


TEKSP.3 Precalculus/Knowledge and Skills. The student uses functions and their properties, tools and technology, to model and solve meaningful problems.

The student is expected to:

P.3A Investigate properties of trigonometric and polynomial functions.

P.3E Solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and

incorporate radian measure where needed.


Lesson Synopsis:

Students review their basic understandings of trigonometry by using sine, cosine, tangent, and the Pythagorean Theorem to solve a variety of right triangle problems.


Vocabulary

• Right Triangle

• Legs

• Hypotenuse

• Pythagorean Theorem

• 30-60-90 Triangle

• 45-45-90 Triangle


What is a right triangle?

It is a triangle which has an angle that is 90 degrees.

The two sides that make up the right angle are called legs.

The side opposite the right angle is the hypotenuse.

leg

leg

hypotenuse

right angle


The Pythagorean Theorem

In a right triangle, if a and b are the measures of the legs and c is the

hypotenuse, then

a2 + b2 = c2.

Note: The hypotenuse, c, is always the longest side.


Find the length of the hypotenuse if

1. a = 12 and b = 16.

122 + 162 = c2

144 + 256 = c2

400 = c2

Take the square root of both sides.

20 = c

2400 c


Find the length of the leg, to the nearest hundredth, if

3. a = 4 and c = 10.42 + b2 = 102

16 + b2 = 100Solve for b.

16 - 16 + b2 = 100 - 16b2 = 84

b = 9.17

2 84b


Find the length of the missing side given a = 4 and c = 5

1. 1

2. 3

3. 6.4

4. 9


Pythagorean Triples


What are Pythagorean Triples?• Three integers that make the

equation a2 + b2 = c2 true are called Pythagorean Triples.

• The numbers 3, 4 and 5 are a very famous Pythagorean Triple.

a2 + b2 = c2

32 + 42 = 52

9 + 16 = 25


Why Memorize Pythagorean Triples?• Remember how much time it took to figure out

8 x 8 before you memorized it?

(8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 64)

• Think of all the work involved to solve this problem:

a2 + b2 = c2

32 + 42 = x2

9 + 16 = x2

25 = x2

5 = x

Wouldn’t it be nice

to just know this is 5?


Good Pythagorean Triples to Memorize:

And multiples of each, like:3x2, 4x2, 5x2

6 8 10


Directions:• The following slides show common

Pythagorean Triples. It is faster to recognize these triples on sight than to apply the Pythagorean Theorem.

• When the right triangle appears, SAY ALOUD the length of the missing side. Try to beat the computer by saying the answer BEFORE it appears in 8 seconds.







x 2 x 2 x 2



x 10 x 10 x 10







x 6 x 6 x 6


Way to Go!!


Lesson 7-3: Special Right Triangles30

Special Right Triangles


Lesson 7-3: Special Right Triangles 31

45°-45°-90° Special Right Triangle

• In a triangle 45°-45°-90° , the hypotenuse is times as long as a leg.

2

2

45°

45°

Hypotenuse

XX

X

Leg

Leg

Example:

45°

45°

5 cm

5 cm

5 cm

2



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

3X

2X

X

Longer Leg

Shorter Leg

Example:

30°

60°

10 cm

5 cm

3

5 cm3



Example: Find the value of a and b.

60°7 cm

a

b

Step 1: Find the missing angle measure. 30°

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°

x

2x3x

a = cm

b = 14 cm

Step 5: Solve for a and b

7 3

Step 4: From the pattern, we know that x = 7 , b = 2x, and a = x .3



Example: Find the value of a and b.

2

45°7 cm

a

b

Step 1: Find the missing angle measure. 45°

45 °

Step 2: Decide which special right triangle applies. 45°-45°-90°

Step 3: Match the 45°-45°-90° pattern with the problem.

45°

45°

x

xx

Step 4: From the pattern, we know that x = 7 , a = x, and b = x .

a = 7 cm

b = 7 cm

Step 5: Solve for a and b

2

2

2


Time for you to practice!Get out your phones!

Go to the Class Website and complete your e-worksheet!

Due Fri Nov 7 (A Day) Due Mon Nov 11 (B Day)

www.themagicofmathryates.weebly.com

Documents

Becky Afghani, LBUSD Math Curriculum Office, 2004 Right Triangles