Pythagorean Theorem

Reporting Category Geometry Topic Working with the Pythagorean Theorem Primary SOL 8.10 The student will a) verify the Pythagorean Theorem; and b) apply the Pythagorean Theorem.


The Pythagorean Theorem
This is not strictly algebra, but it's an interesting cross reference between equation solving and geometry.
On any right-angled triangle, we say that the hypotenuse is the edge facing the 90 • corner. (i.e. the longest edge).

Problem 3:
Determine all possibles values (x, y) such that 2 x+1 + 3 y = 3 y+2 − 2 x The only time that a power of 2 is equal to a power of 3 is 2 0 = 1 = 3 0 . That is, x − 3 = 0 and y − 1 = 0. So x = 3 and y = 1 is the only solution for x, y.

The Pigeonhole Principle
In its strict sense, the Pigeonhole Principle is a combinatorial result. The idea is very simple.
If I have 9 pigeons and only 8 holes to hold them, then at least one of the holes must have more than 1 pigeon, right?
i.e. The Pigeonhole Principle states that given n items and p holes to put them in, where n > p. At least one of the p holes has to have more than 1 item in it.
Problem 4: How many people do you need in a party at minimum to have 2 people born in the same month? 13 Problem 5: 25 students each earn a grade of A, B, or C, the most frequently occurring letter grade be the grade for at least students? 9

Averages are Fun
The average value of n numbers is the sum of the numbers divided by how many numbers there are: 10b + a = 10(7) + 4 = 74 How can you write a 3 digit integer this way?

Problem 8:
A 2-digit number minus 54 equals the 2-digit number but with the digits reversed. Find all possible such 2-digit numbers.
We may rewrite this question as: 10b + a-54=10a Notice that none of b or a can be 0, since that would make either 10b + a or 10a + b a single digit number.
We are looking for pairs of single digits a, b such that b − a = 6. The possible such pairs are (a = 1, b = 7), (a = 2, b = 8), and (a = 3, b = 9) Together We Are Strong! Construction problems usually refer to getting some set amount of work done by a working unit over some amount of time.
Here is what to remember:

Amount per unit × # of units = Total amount
VI Here is what you should NEVER do: Moooooo! If cow-1 is eating the grass, the grass will last cow-1 4 hours. If cow-2 is eating the same patch of grass, the grass will last cow-2 2 hours. If cow-1 and cow-2 both eat the grass, then the grass will last them 3 hours, because 4+2 2 = 3 since cow-1 will eat some and cow-2 will eat some but neither will eat all, so the grass will last their average. This approach is INCORRECT and there is no way to reason behind this approach other than your sheer intuition telling you that you should take their average.

Problem 9:
The cow eating grass problem on the previous scenario with the question being: if cow-1 and cow-2 both eat the grass together, how long will the grass last them?
Let the total amount of grass be n.
Cow-1 can eat n ÷ 4 amount of grass per hour; Cow-2 can eat n ÷ 2 grass per hour.
Then cow-1 and cow-2 combined can eat n 4 + n 2 = 3n 4 amount of grass per hour.
Then the patch of grass n will last cow-1 and cow-2, together, We'll end with a miscellaneous problem: The digits 1, 2, 3, 4, 5 and 6 are each used once to compose a six digit number abcdef such that the three digit number abc is divisible by 4, bcd is divisible by 5, cde is divisible by 3 and def is divisible by 11. What is this number abcdef ? abcdef = 324561 (Hint: Look for where to start on the problem, which number is easy to start tackling down. Try to remember your divisibility rules, and in case you didn't know, a number is divisible by 11 if and only if the sum of its odd-positioned digit minus the sum of its even-positioned digit is divisible by 11. For example, 231 is divisible by 11 because 2 + 1 − 3 = 0 and 0 is divisible by 11; likewise 3927 is divisible by 11 because 7 + 9 − (2 + 3) = 11 which is divisible by 11.)