Post on 23-Dec-2015
Becky Afghani, LBUSD Math Curriculum Office, 2004
Right Triangles
Becky Afghani, LBUSD Math Curriculum Office, 2004
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.
Becky Afghani, LBUSD Math Curriculum Office, 2004
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.
Becky Afghani, LBUSD Math Curriculum Office, 2004
Vocabulary
• Right Triangle
• Legs
• Hypotenuse
• Pythagorean Theorem
• 30-60-90 Triangle
• 45-45-90 Triangle
Becky Afghani, LBUSD Math Curriculum Office, 2004
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
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.
Becky Afghani, LBUSD Math Curriculum Office, 2004
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
Find the length of the missing side given a = 4 and c = 5
1. 1
2. 3
3. 6.4
4. 9
Becky Afghani, LBUSD Math Curriculum Office, 2004
Pythagorean Triples
Becky Afghani, LBUSD Math Curriculum Office, 2004
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
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?
Becky Afghani, LBUSD Math Curriculum Office, 2004
Good Pythagorean Triples to Memorize:
And multiples of each, like:3x2, 4x2, 5x2
6 8 10
Becky Afghani, LBUSD Math Curriculum Office, 2004
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.
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
x 2 x 2 x 2
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
x 10 x 10 x 10
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
Becky Afghani, LBUSD Math Curriculum Office, 2004
x 6 x 6 x 6
Becky Afghani, LBUSD Math Curriculum Office, 2004
Way to Go!!
Becky Afghani, LBUSD Math Curriculum Office, 2004
Lesson 7-3: Special Right Triangles30
Special Right Triangles
Becky Afghani, LBUSD Math Curriculum Office, 2004
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
Lesson 7-3: Special Right Triangles 32
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
Lesson 7-3: Special Right Triangles 33
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
Lesson 7-3: Special Right Triangles 34
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
Becky Afghani, LBUSD Math Curriculum Office, 2004
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