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Who was Pythagoras, and what did he contribute to geometry?

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Explain how geometry is used in your everyday life.

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The word geometry comes from two Greek words, geo, which means the earth, and metron, which

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means to measure.

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Today, geometry is more the study of shapes than it is the study of the earth.

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Basically, geometry is the branch of mathematics that deals with the position, the size, and

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the shape of figures.

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One of the greatest mathematicians was an ancient Greek named Pythagoras.

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He discovered some of the most important mathematical concepts that came to be called geometry.

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One observation he made was that gravity is vertical, or 90 degrees to the horizon.

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From this observation, Pythagoras discovered that the 90 degree angles from four right

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sided triangles make up a square.

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Watch this.

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If I have one right angle, and I place three other right angles around it, like this, I

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eventually wind up with, ta-da, a square.

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That's pretty neat.

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Let's do the math.

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Knowing what Pythagoras discovered about the right angle, can you calculate how many degrees

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are in this square?

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If you multiply 90 degrees times four, you're right.

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This square has 360 degrees.

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What other shape has 360 degrees?

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A circle.

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You know, Pythagoras proved that there are relationships between different geometric

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shapes.

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What relationships can you see between other geometric shapes?

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Get this.

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Pythagoras found out even more laws about the right triangle.

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If we look at the same square, but just a little differently, we can see that half the

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area of this square equals a right triangle.

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Now, how can we use math to calculate the remaining angles of a right triangle?

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Simple.

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Squares are 360 degrees, we know this.

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If we divide it in half, this triangle must equal 180 degrees.

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Now, we know this is a right triangle.

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This equals 90 degrees.

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If we subtract that from 180, we get 90 degrees.

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These two angles must add up to 90 degrees.

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This is true for every right triangle.

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It's true for this right triangle, it's true for this right triangle, and it's even true

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for right triangles that look like this.

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In order to calculate the remaining angles of a right triangle, you have to use math

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and geometry.

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Geometry is used in everything we do, from constructing roads and buildings to playing

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football or pool.

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Okay, here's a big play.

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It's you and me, okay?

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I'll toss the big pass to you, you go down and out, got it?

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Got it!

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Great!

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Now, let's see.

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If I toss the ball directly to Jennifer and don't anticipate where she'll be, I'll miss

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her completely.

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However, if I know she's cutting right, and I throw the ball at the correct angle, I should

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get the ball to her.

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Hey!

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My perfect pass just created a right triangle.

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Geometry is everywhere.

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Hey, way to go, Van!

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Without geometry, it would be impossible to organize precise patterns and play a simple

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game of football.

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My friend Lynn Chappell is an 8th grade math teacher at Huntington Middle School in Newport

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News, Virginia.

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Let's see what information she has about Pythagoras and geometry.

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The most important discovery that Pythagoras made was the relationship between the longest

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side of a right triangle and the two shorter sides.

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The longest side of the right triangle is called the hypotenuse.

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Now, remember that Pythagoras' theorem is A squared plus B squared equals C squared.

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Now, who could tell me what that means?

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Charmaine.

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The sum of the squares of the two shorter sides, A plus B, equals the square of the

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longest side, C, which is the hypotenuse.

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Good answer.

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Now, we're going to mark the right triangle that we have on this paper, and the shorter

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sides, also called the legs, are A and B, and the longest side is C, and remember we

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call that the hypotenuse.

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Now, what Pythagoras did was draw a square on the side of A, and remember that a square

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is a number times itself, A times A, and he drew a square on the side of B, B times B,

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and he drew a square on the side of C, C times C.

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What we're going to do is we're going to cut A squared off of the side, and then we're

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going to cut B squared and make them fit into C squared to prove that Pythagoras was right.

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First take your straight edge, and we're going to draw some parts of B so that we can cut

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it and it will fit.

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On the long side of C, come straight down through B squared until you touch the edge.

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Now connect the lower corner of B to the bottom edge of A squared.

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This will form a perpendicular line.

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Now take your scissors and cut out A squared in one piece and B squared in the pieces that

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you've cut it into, and then we'll fit it all on to C squared to prove that Pythagoras

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was right.

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Well, have all of you fit your pieces together?

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Yes.

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Then I guess Pythagoras was right.

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And you know, Pythagoras also believed or postulated that the shortest distance between

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two points is a straight line.

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Well, how come if you throw a ball from point A to point B, then it curves or arcs?

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Well, Van, that's rather very simple.

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Ever heard of something called gravity?

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Yeah, I've heard of gravity!

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In 1600, Johannes Kepler, a famous astronomer, proved that the planets orbited the sun in

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an ellipse.

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That's another geometric shape.

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If you take a circle and squash it a bit, you get an ellipse.

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Like our football example, if we want to navigate from Earth to Mars, we have to take into account

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where Mars will be within its elliptical orbit.

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What information did scientists first discover about Mars?

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Humans have known of Mars since before recorded history.

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In 1609, a man by the name of Galileo first viewed Mars through his newly invented telescope.

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Although his telescope was no better than a modern toy, it revealed enough to prove

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that Mars was a large sphere, a world shaped like the Earth.

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Could this other world be inhabited?

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Besides using the telescope, how else do scientists collect information on Mars?

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Let me tell you!

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NASA's Mariner 4 was the first spacecraft to take close-up pictures of the red planet.

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As it flew past Mars in 1965, it showed a heavily crated surface.

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Six years later, in 1971, Mariner 9 arrived at Mars and became the first artificial object

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ever to orbit another planet.

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Mariner 9 saw the Valles Marineris, a canyon that stretches 4,500 kilometers, or 2,800

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miles, across the face of Mars.

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It is so long that if it were on Earth, it would stretch all the way from Los Angeles,

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California to New York, New York.

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All these discoveries by Mariner were seen from above the surface of Mars.

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What we really needed was a view from the Martian surface.