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SAT Math 4.4 Geometry and Measurement 210 Views


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Description:

SAT Math 4.4 Geometry and Measurement

Language:
English Language

Transcript

00:02

Here’s your shmoop du jour, brought to you by radius envy.

00:06

As observed by the renowned psychoanalyst Segment Freud.

00:13

If the large circle has a center of H and the small circle has a center of I,

00:18

and K is the midpoint of IH, what percent of the large circle (Circle H) is shaded?

00:25

And here are the potential answers...

00:29

We’ve got circles and triangles and intersections and midpoints up the wazoo.

00:34

That shaded area is what we’re after…

00:36

it’s both part of a triangle and part of both circles, so…where to start?

00:41

Well, what are we given? We’re told that segment KI is half the length of HI.

00:46

We aren’t given any numbers, so… heck, let’s just throw some in here.

00:50

If we call KI “1,” then HI can be “2.”

00:55

We’re also told that I is the midpoint of the smaller circle, so we know KI is the radius

01:00

of that smaller circle. So is IL…which makes that segment also equal to “1.”

01:06

Now…what about the radii of the larger circle?

01:09

We’ve already determined that HI is 2… so HJ, another radius, would also be 2.

01:14

And we know that angle IHJ is 90 degrees.

01:17

Well, now we’ve got a right triangle with 2 sides of equal length…meaning we have an

01:21

isosceles triangle on our hands…

01:23

…which, by definition, has two angles of 45 degrees.

01:26

All right, so angle KLI is 45 degrees.

01:29

A full circle is 360 degrees… so the area of the shaded section is 45/360ths of the smaller circle.

01:37

45 divided by 360 is 0.125…or 12.5% of the area of the small circle.

01:43

12.5% is one of our answer choices… but don’t let ‘em rope you in here.

01:48

We’re looking for the percentage of the larger circle.

01:51

The area of the small circle…or, of any circle, for that matter, is πr squared.

01:56

We’ve decided to call our radius “1,” so our area in this case is π(1)2, or just…pi.

02:03

The shaded area is therefore 12.5% of pi, or 0.125 pi.

02:08

So…what percentage of the larger circle is that?

02:11

Well, we need to do the same thing we just did to find the area of the big guy.

02:15

Here, our radius is 2, so the area is pi times 2 squared… or 4 pi.

02:20

So finally, to find the percentage, we take our 0.125 pi divided by 4 pi…

02:26

…to get 0.03125…

02:28

…or 3.125%.

02:30

Answer B.

02:32

That’s a long way to go just to find a tiny shady area.

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