Download MATH 5: HOMEWORK 11 Square Root The square

MATH 5: HOMEWORK 11
APRIL 17, 2011
Square Root
√
√
The square root of a number a is written a and is a number whose square
is a. For example: 25 = 5
√
√
√
because 52 = 25. When multiplying two square
root numbers
we have: a b= ab
√
√ √
but if we add two square root numbers: a+ b is not a + b.
Pythagorean Theorem
In a right triangle with legs a,b and hypotenuse c,
a2 + b2 = c2 , c =
p
a2 + b2
A proof of this theorem:
In this square, the total area is
(a + b)2 = (a + b)(a + b) = a(a + b) + b(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2
The area of each triangle is 21 ab and the area of the shaded square is c2 . Thus, we get
1
a2 + 2ab + b2 = 4 ab + c2
2
which gives
a2 + b2 = c2
√
For example, in a square with side 1, the diagonal has length 2. It is possible to find a right triangle where
all sides are whole numbers. One such triangle has sides 3, 4, 5.
Homework
1. What is
2.
3.
4.
5.
6.
√
16 =
√
81 =
p
10, 000 =
√
108 =
√
50 =
Write the squares of the first 20 numbers.
If, in a right triangle, one leg has length 1 and the hypotenuse has length 2,q
what q
is the other leg?
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1
6
2
Find 2 7 (hint: you do not need to compute what this number itself is); 16 ; 49
Can you find a right triangle with all sides whole numbers and the hypotenuse 13?
Find the height and area of the figure below. The lengths of three sides are given, the 2 marked
angles are right angles.
7. Simplify:
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72 =;
√
75 =;
√
√
√
24 6 =; 4500 =