Download 20 Pythagorean Theorem1

Pythagorean Theorem I
If you know the lengths of two sides of a right triangle, a special formula can be used to find the
length of the third side. The hypotenuse of a right triangle is the side opposite the right angle.
The hypotenuse is longer than either of the two other sides called legs.
a2 + b 2 = c 2
The number of square units in the square on
the hypotenuse, c, is equal to the sum of the
number of square units in the squares on
both legs, a and b.
The Pythagorean Theorem also allows you to determine if a triangle is a right triangle or not.
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If the sides of a triangle satisfy the Pythagorean Theorem, then the triangle is a right
triangle.
If the sides of a triangle do not satisfy the Pythagorean Theorem, then the triangle is not a
right triangle.
Pythagorean Triples (Pythagorean Triad)
Most right triangles do not have all sides that can be measured in whole numbers. A
"Pythagorean Triple" is a set of positive integers, a, b and c that fits the above rule. Two
common Pythagorean Triples are: 3:4:5 and 5:12:13. These ratios and any numbers directly
proportional to them are likely to appear on tests.
Complete the chart below to find other sets of numbers that are proportional to these triples.
Side a = 3
Side b = 4
Side c (hypotenuse) = 5
6
8
10
30
40
50
15
20
25
Side a = 5
10
500
50
Side b = 12
24
1200
120
Side c (hypotenuse) = 13
26
1300
130
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Converse of Pythagoras’ theorem: If c2 = a2 + b2 then ∠C is a right angle (where c is the length
of the side opposite angle C).
Irrational Numbers
Consider the sequence
This sequence of positive real numbers is strictly increasing and
is a whole number if and
only if n is a perfect square such as 36 or 49. The sequence tends to infinity, that is, there is
no upper bound for
.
A right-angled triangle with equal side lengths is an isosceles triangle. Hence the angles are
45°, 45° and 90°. If the length of a side is 1 then the hypotenuse is of length
(since 12 + 12
= 2).
Example: Find the length, correct to 2 decimal places, of
the missing side in the right triangle opposite.
1. Find the length of the third side of each right triangle.
29
10√2
5
2
Special right triangles
45°-45°-90° triangle
2 ≈ 1.414
30°-60°-90° triangle
3 ≈ 1.732
2. Which is closest to the length of x in the 45°-45°-90° triangle below?
A. 12
B. 13
C. 6
D. 7
3. Which is closest to the length of x in the 30°-60°-90° triangle below?
A. 13
B. 19
C. 23
D. 26
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4. Which of the triangles below are right-angled triangles? Name the right angle in each
case.
Triangle ABC is not a right-angled triangle since 142 + 272 ≠ 322.
Triangle DEF is a right-angled triangle since 152 + 362 = 392. ∠F is the right angle
5. For ∆ABC
a. Find the length of AC.
b. Find the area of ∆ABC.
92 + AC2 = 412 = 1681
AC2 = 1681 – 81 = 1600
AC = 40
Area: ½ 40×9 = 180 cm2
6. A ladder 13 feet long leans against a wall. If the foot of the ladder is 5 feet from the wall,
how far up the wall does the ladder reach?
52 + b2 = 132 = 169
b2 = 169 – 25 = 144
b= 144 =12
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