Alexandra's Pythagoras Growing Post

Pythagorean Triple
The Pythagorean Triple is a relation among the three sides of a right triangle.
The theorem is as follows:
The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation:

Using the perfect square chart from one to ten, I found out that I can use 6,8, & 10 to make a Pythagorean Triple. Here's what I did.

I used 6 as my a and I used 8 as my b.

a² + b² = c²
36 + 64 = 100
6² + 8² = 10²

Click here to see how to find the missing leg of a right triangle.
Click here to see how to find the missing hypotenuse of a right triangle.

You're locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window?





My own word problem.
Two joggers run 8 miles north and then 5 miles west. They discovered that there is another way to go back to their starting point. They went from west straight to north without turning. How many miles did they travel from west back to north?

Kadille Dave's Pythagoras Growing Post

1) A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime. The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution. The converse is not true. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 don't have an integer common multiple because √2 is irrational.

If (a, b, c) is a Pythagorean triple,is (ka, kb, kc) for any positive integer k. The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution.

a=3x3
b=4x4
c=5x5

a+b=c

9+16=25

2)Find The Missing Side


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3) School Problem Mr. H jumped into a canoe to cross a river. That runs east to west and is 20m wide. The current carried him downstream 8m before he reach the other side. How did he travel in total?

Given: 20m= edge to edge of the river
8m= downstream effect
c= mr. H travelled






This looks how the problem talks about.









solution on how to solve the problem











4) Own Problem

Angelo is planning to slide on a mountain that has a measurement of 156 meters from end to end. There is a building at the summit 10 meters wide and eight meters high. The center of the building is standing exactly on the summit of the building. The summit of the mountain is in the center of the mountain. If someone travel from the summit to the base of the mountain, he will travel 100 metres. The starting point is where the edges of the building is. Find how high did Angelo start sliding?

This diagram shows the labels of sides or measurements in the problem.

The real missing side is from the starting point down to the base of the mountain.
-but the other sides of the triangle are not the exact ones
-there is a building on top of the mountain

letter points
z-distance from the summit to the end of the mountain












Z-the distance from the starting point to the ground that is measured using the slope.











c-distance from the summit to the edge of the mountain.











H - the height of the mountain from the summit down to the base











h- the distance on how high Angelo started sliding from the ground











Q- the distance of the edges of the mountain divided into 2 minus 5









P=Starting Points












plan:
to solve the exact missing sides using: ratio and proportion, subtracting, dividing, and finding the missing side

1. Ratio and Proportion - solve for c
The angle of the small triangle inside the building has the same angle as the whole triangle from one end
















H - the height of the mountain from the summit down to the base
it has a measurement of 62.58 metres













2. Solve for z










3. Solve for Q











Lastly, solve the problem. Because we got the exact measurement of the given sides

a-h(?)
b-q
c-Z

Solution
(click the picture to enlarge)

Nadine's Pythagoras Growing Post

Part 1
A Pythagorean Triple is right triangle that has three integers ( a = leg, b = leg, and c = hypotenuse) and uses a set of numbers (1,2,3..etc. - no decimal) that satisfy a² + b² = c². (you need to label it a, b, and c. you can tell which is the hypotenuse because it is always opposite the right angle.)
Example :5² + 12² = 13² (5:12:13) && 6² + 8² = 10² (6:8:10)

Part 2

Part 3
Lefrog is standing in a room that has tables that are 120 cm tall and chairs that are 1.4 tall, the entrance doorway to the room is 2 m high. Lefrog is 3 m away from the doorway. If Lefrog's tongue leaves his mouth at the same height as a table, how far does tongue have to stretch to eat the fly?

Part 4
The merchants often fashioned tents out of a piece of cloth thrown over tent poles and then secured to the ground with stakes forming an isosceles triangle. How long would the cloth have to be so that the opening of the tent was 4 meters high and 3 meters wide?

Mariya's Pythagoras Growing Post

Part I:
A pythagorean triple is a right angle triangle that has three integers (which are a, b, and c.) that satisfy a² + b² = c².
An easy example is 3² + 4² = 5², but, the example I'm going to use is 6² + 8² = 10².





another way to show 6² + 8² = 10² :

Part II:

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Part III:
You've just picked up a ground ball at first base, and you see the other team's player runnig towards thrid base.
How far do you have to throw the ball to get it from first base to thrid base, and runner out?

Part IV:
Danny's dog house is shaped like a tent. The slanted sides are both 5 feet long and the bottom of the house is 6 feet across. What is the height of this dog house, in feet, at its tallest point?