a closer look at right triangles
DESCRIPTION
A closer look at right triangles. leg ² + leg² = hypotenuse². Acute angles are complementary. Two acute angles. The Right Triangle. hypotenuse. leg. One right angle. leg. Pythagorean Theorem. The Pythagorean Theorem describes the relationship between the sides of a right triangle. - PowerPoint PPT PresentationTRANSCRIPT
One right angle
Two acute angleshypotenuse
leg
leg
leg² + leg² = hypotenuse²
Acute angles are complementary.
The Pythagorean Theorem describes the relationship between the sides of a right triangle.
leg² + leg² = hypotenuse²
A Pythagorean triple is a set of integers, a, b, and c, that could be the sides of a right triangle if a² + b² = c².
3, 4 and 5 are a Pythagorean triple because 3² + 4² = 5² and all three numbers are whole numbers.7, 8 and 12 are NOT a Pythagorean triple because 7² + 8² = 12² even though they are all whole numbers.5, 9.5 and √115.25 are NOT a Pythagorean triple - 5² + 9.5² = √115.25 ² BUT the three numbers are not whole numbers.
Many mathematicians over the centuries have developed formulas for generating side lengths for right triangles. Some generate Pythagorean triples, others just generate the side lengths for a right triangle.
Masères
n n² - 1 2
n² + 1 2
, ,
Use Pythagoras’ formula to find a Pythagorean triple when n is an odd number.
Find a Pythagorean triple using Pythagoras’ formula when n = 7.
aa²4
, ,- 1 a²4
+ 1
Use Plato’s formula to find a Pythagorean triple when a is an even positive integer greater than 2.
Find a Pythagorean triple using Plato’s formula when a = 6.
xyx - y 2
x + y 2
, ,
Euclid’s formula won’t always give you a Pythagorean triple. If you restrict values of x and y to either two even or two odd numbers in Euclid’s formula you will at least have two whole numbers.
Find the lengths of the sides of a right triangle if x = 7 and y = 9.
2pq p² - q², , p² + q²
Maseres wrote many mathematical works which show a complete lack of creative ability. He rejected negative numbers and that part of algebra which is not arithmetic. It is probable that Maseres rejected all mathematics which he could not understand.
Masères
2pq
p² - q²
p² + q²Use Maseres’ formula to find a Pythagorean triple when you are given two whole numbers.
Find a Pythagorean triple using the numbers 9 and 10.
10
10
This is a special right triangle called a 45°-45°-90° triangle. Why is it given this name?
45°
45°
This is a special right triangle because there is a special relationship between the sides of the triangle.
Find the length of the hypotenuse of this triangle. Simplify the radical.10√2
Check out these other 45°-45°-90° triangles. Find the length of the missing side – simplify radicals. Can you figure
out the special relationship between the sides of a 45°-45°-90° triangle?
4
4
45°
45°
4√2
7
7 7√2
9
99√2
12
1212√2
2
2
45°
45°
2√2
1
11√2
leg
leg
What is the special relationship between the lengths of the sides of a 45°-45°-90° triangle?
45°
45°
What is the length of each missing side of this triangle?
leg√2
2010
This is a special right triangle called a 30°-60°-90° triangle. Why is it given this name?
60°
30°
This is a special right triangle because there is a special relationship between the sides of the triangle.
Find the length of the missing leg of this triangle. Simplify the radical.
The legs of this type of triangle are given special names.
10√3
Check out these other 30°-60°-90° triangles. Find the length of the missing side – simplify radicals. Can you figure
out the special relationship between the sides of a 30°-60°-90° triangle?
84 60°
30°4√3
147
7√3
9√3
918
12√3
12 244 2
60°
30°2√3
21
1√3
60°
30°
60°
30°
60°
30°
60°
30°
2 (short leg)shortleg
What is the special relationship between the lengths of the sides of a 30°-60°-90° triangle?
60°
30°
What is the length of each missing side of this triangle?
short leg√3
Pythagorean Triples and Special Right TrianglesPythagorean Triple - A set of three whole numbers such that a² + b² = c²
Pythagoras’ formula Plato’s formula
Euclid’s formula Maseres’ formula
n n² - 1 2
n² + 1 2, , a
a²4 , ,- 1
a²4 + 1
xyx - y 2
x + y 2
, , 2pq p² - q², , p² + q²
leg
leg45°
45°
leg√22 (short leg)short
leg
short leg√3
60°
30°
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