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Right and Special Right Triangles, a la Shmoop. Put down your chainsaws...

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...because the annual Lumberjack Ball is tonight! Unfortunately, Lumberjack Bill, the one who

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can never get a date...

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...decided to steal the lumberjack dancin' square the night before...

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...leaving the poor lumberjacks to rebuild the whole thing before tonight's ball.

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Here's what the square looked like last night.

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The lumberjack waltz requires there to be a distance of 4 between point B and point

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D.

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Knowing length B prime D prime is 4, what is the length of a side of the square ABCD?

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Here are your choices: Well, to start, it helps to know that squares

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have four right angles and four equal sides.

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Since angle D is in the corner of square ABCD, we know it's a 90 degree angle.

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Zoom in on triangle B prime D D prime. We know it's a right triangle, so we can use

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the Pythagorean theorem to solve it.

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We know its hypotenuse, but not its two side lengths.

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Whatever will the lumberjacks do?

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Since A prime B prime C prime D prime is also a square, we know that angle D prime is also

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90 degrees.

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Angle D D prime C is a straight angle, so it has a measure of 180 degrees.

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Subtract 90 degrees for angle B prime D prime C prime, and we only have 90 degrees to share

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among the two smaller side angles.

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Splitting them up evenly, each angle gets 45 degrees.

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If we do that with all the angles, we'll see that our triangle B prime D D prime is a special

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right triangle, a 45-45-90 triangle.

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Since both its acute angles are congruent, we know the lengths of its legs are congruent.

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Good thing, too; otherwise square dancing would be a big mistake.

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Now we can use the Pythagorean theorem and replace both a and b with the same length:

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x.

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That's the wonderful thing about 45-45-90 triangles. If the length of the leg is x,

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the hypotenuse will always be equal to x times the square root of 2.

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We're looking for the side of the big square, ABCD.

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If we look at the picture, we know that D prime D is the same length as D prime C, so

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we just have to multiply 2 root 2 by 2. That gives us 4 root 2.

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So, long story short... if it isn't already too late... our answer is D.

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Now those lumberjacks can rebuild their perfect square and get ready to dance.

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Swing your chainsaw 'round and 'round!

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