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A Simple But Common Application of Quadratic Equations

Introduction: Anyone that builds anything with a 90 degree angle will sooner or later need to use the Pythagorean Theorem which states: For any right triangle with sides of length a, b, and c, a2+b2=c2 with c being the longest side or hypotenuse.

Example: I encountered a problem very similar to this when building a roof on a greenhouse. The dimensions consist of a rafter length c=7.5ft (so I can use 8ft lumber for my rafters) and height a=1ft as shown below.  What will be the width of the greenhouse W?

Here, we need to let W/2 = b  where "b" is the length of the bottom side of the two equivalent right triangles. We get the equation  12 + b2 = 7.52, a quadratic equation in b.

Applying the addition property of equality and subtracting 12 from both sides results in b2 = 7.52 - 12 or  b2 = 55.25 .

Extracting square roots from both sides results in b = 7.433 ft (rounded).  Since 2b = W, W=14.87 ft (rounded).  We can convert the 0.87ft into inches by multiplying 0.87ft by 12in/ft to get 10.44 inches (rounded). You can convert the 0.44 inches to 1/16ths of an inch by multiplying 0.44 in by (16 sixteenths/1 in) to get 7/16.

So the total width is W= 14feet, 10 7/16 inches.

More Examples I Found

Waybuilder.net  - Determine boundaries of building layout by using the Pythagorean Theorem to calculate the diagonals.

Contractortalk.com - "I need help with this question on roof framing."

Lowes.com - Finding squares on large projects.

Servicemagic.com - Framing rake walls.


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