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7-2 Notes on Pythagorean Theorem
Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of
the legs.
Please note the following:
 Pythagorean Theorem only applies to RIGHT triangles
 We know that the HYPOTENUSE of a right triangle is (1) longer than the
other two sides and (2) is opposite the right angle
 Both of the non-hypotenuse sides are called the legs
 The converse of the Pythagorean Theorem also holds. You can use this
theorem to help you determine whether a triangle is a right triangle given
the measures of all three sides.
 You can also use side lengths to classify a triangle as acute or obtuse:
 If a2 + b2 > c2 (where c is the longest side), then the triangle is acute.
 If a2 + b2 < c2 (where c is the longest side), then the triangle is obtuse.
Find the missing lengths.
1.
2.
3. Determine whether each set of numbers can be the measures of the sides of a triangle. If so, classify the
triangle as acute, right, or obtuse.
a) 11, 60, 61
b) 2 3, 4 2,3 5
c) 6.2, 13.8, 20
Pythagorean Triples
A set of three integers that can be the lengths of the sides of a right triangle is a
Pythagorean triple.
 If the measures of the sides of any right triangle are NOT whole numbers, then
the measures do NOT form a Pythagorean triple.
 If a, b, and c is a Pythagorean triple, then so is ak, bk, and ck where k is a
nonzero whole number.
 The simplest Pythagorean triple is the set “3, 4, 5.” These numbers are the
lengths of the sides of a “3-4-5” Pythagorean right triangle.
The following list contains all of the Pythagorean triples in which no number is more
than 50:
3, 4, 5
9, 12, 15
14, 48, 50
20, 21, 29
5, 12, 13
9, 40, 41
15, 20, 25
21, 28, 35
6, 8, 10
10, 24, 26
15, 36, 39
24, 32, 40
7, 24, 25
12, 16, 20
16, 30, 34
27, 36, 45
8, 15, 17
12, 35, 37
18, 24, 30
30, 40, 50
4. Verify that 21, 72, 75 is a Pythagorean triple.
5. Fred is locked out of his house. The only open window is on the second floor, which is
12 feet above the ground. He needs to borrow a ladder from his neighbor. If he must
place the ladder 5 feet from the house to avoid some bushes, what length of ladder does
Fred need?
6. Determine whether XYZ is an acute, right, or obtuse triangle for the given vertices.
Explain.
a) X(-7, -3), Y(-2, -5), Z(-4, -1)
b) X(1, 2), Y(4, 6), Z(6, 6)
7. Find the perimeter of the trapezoid:
8. The sides of a triangle have lengths x, x+5, and 25. If the length of the longest side is
25, what values of x makes the triangle a right triangle?
9. The sides of a triangle have lengths 2x, 8, and 12. If the length of the longest side is
2x, what values of x make the triangle acute?
10. The screen aspect ratio, or the ratio of the width to the height, of a high-definition
television is 16:9. The size of a television is given by the diagonal distance across the
screen. If an HDTV is 41 inches wide, what is the screen size?
11. What is the geometric mean of 45 and 5?
12. Find x, y, and z.