Special Right Triangles
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Here's what the square looked like last night.
The lumberjack waltz requires there to be a distance of 4 between point B and point
Knowing length B prime D prime is 4, what is the length of a side of the square ABCD?
Here are your choices: Well, to start, it helps to know that squares
have four right angles and four equal sides.
Since angle D is in the corner of square ABCD, we know it's a 90 degree angle.
Zoom in on triangle B prime D D prime. We know it's a right triangle, so we can use
the Pythagorean theorem to solve it.
We know its hypotenuse, but not its two side lengths.
Whatever will the lumberjacks do?
Since A prime B prime C prime D prime is also a square, we know that angle D prime is also
Angle D D prime C is a straight angle, so it has a measure of 180 degrees.
Subtract 90 degrees for angle B prime D prime C prime, and we only have 90 degrees to share
among the two smaller side angles.
Splitting them up evenly, each angle gets 45 degrees.
If we do that with all the angles, we'll see that our triangle B prime D D prime is a special
right triangle, a 45-45-90 triangle.
Since both its acute angles are congruent, we know the lengths of its legs are congruent.
Good thing, too; otherwise square dancing would be a big mistake.
Now we can use the Pythagorean theorem and replace both a and b with the same length:
That's the wonderful thing about 45-45-90 triangles. If the length of the leg is x,
the hypotenuse will always be equal to x times the square root of 2.
We're looking for the side of the big square, ABCD.
If we look at the picture, we know that D prime D is the same length as D prime C, so
we just have to multiply 2 root 2 by 2. That gives us 4 root 2.
So, long story short... if it isn't already too late... our answer is D.
Now those lumberjacks can rebuild their perfect square and get ready to dance.
Swing your chainsaw 'round and 'round!