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Geometry and Measurement Videos 48 videos

SAT Math 2.1 Geometry and Measurement
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SAT Math 7.3 Geometry and Measurement 183 Views


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SAT Math 7.3 Geometry and Measurement


Transcript

00:02

Here’s your shmoop du jour, brought to you by squares within squares.

00:07

It’s like a spiraling S’more vortex.

00:10

ABCD is a square.

00:12

Points A prime, B prime, C prime, and D prime are constructed by translating the points

00:17

A, B, C and D over two-thirds the length of the side.

00:23

If the perimeter of A prime B prime C prime D prime is

00:28

8 times the square root of 5, what is the perimeter of ABCD?

00:33

Here are the potential answers...

00:37

We’ve got one square – ABCD. Inside that square, we’ve got a slightly smaller one…

00:43

that we’ve created by plunking down four new points along the first square’s edges.

00:48

There are really only two juicy bits of information we’re given.

00:52

First, that the new points, which we call “prime,” are translated from their original

00:58

locations 2/3 of the way along each side.

01:01

And second, that the perimeter of that inner circle is 8 square root of 5.

01:07

All righty…the easiest piece to extrapolate is the perimeter of the inner square.

01:12

We know that square has 4 equal sides…so if the total perimeter is 8 square root of

01:17

5, then the length of each side must be exactly 1/4 of that…or 2 square root of 5.

01:23

The next thing we should notice is that our inner square has magically created some triangles…

01:29

We’re not given any measurements for the outer square, but we can always call one

01:33

side “x”, so… let’s do that.

01:36

Because we’re told where all those “prime” points are, we know that one side of our triangle

01:40

is going to be 1/3x and the other side is going to be 2/3x.

01:44

We now have ways to reference all three sides of one of our triangles.

01:48

So…let’s Pythagorize ‘em…

01:50

1/3x squared plus 2/3x squared equals 2 square root of 5 squared.

01:57

We get 1/9x squared plus 4/9x squared equals 4 times 5.

02:02

Simplifying further, 5/9x squared equals 20.

02:07

We can now multiply both sides by 9 to get 5x squared equals 180…

02:11

…and then divide both sides by 5 to get x squared equals 36.

02:15

Last step – take the square root of both sides, and x equals 6.

02:19

Now…be careful – that’s only the length of one side, and we’re looking for the entire perimeter.

02:25

So…just multiply that 6 by 4 and we get there. The answer is 24 or E. Done.

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