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Geometry Students…
Welcome back from spring break! The rest of
the year will FLY by!
This lesson includes a review of the Pythagorean Theorem. Put
the PPT on “play” and take notes on this information. Then read
page 417 to page 419 in your text book
Do the homework to the best of your ability.
Then, open PPT 8.2 and check your work.
Chapter 8.1
Pythagorean Theorem and its Converse
Remember that the distance formula is based off of the
Pythagorean Theorem.
𝑑=
(𝑥2 − 𝑥1 )2 +(𝑦2 − 𝑦1 )2
Complete the list of perfect squares:
1
1
9
17
2
4
10
18
3
9
11
19
4
12
20
5
13
21
6
14
22
7
15
23
8
16
24
Pythagorean Theorem: 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.
2
2
𝑎 +𝑏 =𝑐
2
Hypotenuse – longest side of a right triangle.
“c” always represents the _____________________
Pythagorean Triple: a set of nonzero whole numbers
a, b and c that satisfy the equation:
In other words, if there are no decimals or
fractions for any of the three sides, it is a “triple”
2
2
𝑎 +𝑏 =𝑐
2
This triangle is a Pythagorean triple,
Since all sides are whole numbers.
Are either of the triangles a
Pythagorean Triple?
𝑎2 + 𝑏2 = 𝑐 2
?
5 + 12 = 132
2
2
Yes! This is a
Pythagorean triple!
25 + 144 = 169
𝑎2 + 𝑏2 = 𝑐 2
82
+
152
?
= 172
64 + 225 = 289
Yes! This is also a
Pythagorean triple!
Is the triangle a
Pythagorean Triple?
𝑎2 + 𝑏 2 = 𝑐 2
2
2
20 + 21 = 𝑐
20
Yes!
2
400 + 441 = 𝑐
841 = 𝑐 2
21
841 =
29 = 𝑐
𝑐2
2
Using simplest radical form:
82 + 𝑥 2 = 202
2
64 + 𝑥 = 400
𝑥 2 = 336
20
𝑥 2 = 336
𝑥 = 16 ∙ 21
8
𝑥
𝑥 = 4 21
This is NOT a Pythagorean triple.
How to determine if a triangle is right, acute, or obtuse
by its side lengths.
𝑎2 + 𝑏 2 = 𝑐 2 ONLY for a right triangle.
2
2
2
20 + 21 = 29
29
20
21
If this is true, then this MUST be a
right triangle.
2
2
2
But what if 𝑎 + 𝑏 = 𝑐 is not true?
First, place all three numbers in the formula:
remember “c” is the longest side!
?
202 + 212 = 312
31
Next square each number.
?
400 + 441 = 961
20
?
21
841 = 961
If the hypotenuse is longer, the
legs must be widened, meaning it
is an obtuse triangle.
If the hypotenuse is too short, the
legs must be pulled together,
meaning it is an acute triangle.
Converse of the Pythagorean Theorem: if the square
of the length of one side of a triangle is equal to the
sum of the squares of the lengths of the other two
sides, then the triangle is a right triangle.
2
𝑎
+
2
𝑏
=
2
𝑐
If the square of the length of the longest side of a
triangle is greater than the sum of the squares of the
lengths of the other two sides, then the triangle is a
an obtuse triangle.
2
𝑐
>
2
𝑎
+
2
𝑏
(obtuse)
If the square of the length of the longest side of a
triangle is less than the sum of the squares of the
lengths of the other two sides, then the triangle is a
an acute triangle.
2
𝑐
<
2
𝑎
+
2
𝑏
(acute)
Classify a triangle as right, acute or obtuse:
Side Lengths: 6, 11, and 14
142 __62 + 112
𝑜𝑏𝑡𝑢𝑠𝑒
Side Lengths: 7, 8, and 9
Side Lengths: 6, 8, and 10
92 __82 + 72
102 __82 + 62
𝑎𝑐𝑢𝑡𝑒
𝑟𝑖𝑔ℎ𝑡
𝑐 2 = 𝑎2 + 𝑏 2 (right)
𝑐 2 < 𝑎2 + 𝑏 2 (acute)
𝑐 2 > 𝑎2 + 𝑏 2 (obtuse)
Homework – First read page 417-419 in your text book. Page 420-421: 1-11 odd, 18, 21, 24, 25, 32