SAT Math 7.3 Geometry and Measurement

Polygons | Perimeter |

Problem Solving and Data Analysis | Center, shape, and spread |

Product Type | SAT Math |

Quadrilaterals | Squares Squares and Rectangles |

SAT Math | Geometry and Measurement |

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

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

Here are the potential answers...

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

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

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

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

locations 2/3 of the way along each side.

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

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

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

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

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

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

side “x”, so… let’s do that.

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

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

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

So…let’s Pythagorize ‘em…

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

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

Simplifying further, 5/9x squared equals 20.

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

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

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

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

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