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A pole vault stick is leaning against a wall, touching the wall 16 ft above the ground. The bottom end of the stick is 12 ft from the wall. How many feet long is the stick?

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Question:

A
20

explanation

The situation described will give the figure of a right triangle with the vertical line measuring 16 feet, the horizontal line measuring 12 feet, and the hypotenuse (or the diagonal line) of unknown length.

When you see a right-triangle type of problem, check first to see if it can be solved using the many special properties of right triangles. This will simplify computation. For example, the Pythagorean theorem can be used in this case:

\( 12^2+ 16^2= c^2\)

\(144+256=c^2\)

\(400=c^2\)

\(\sqrt{400} = \sqrt{c^2} \)

\(20 = c\)

TIP: Take note that the two perpendicular sides measuring 12 and 16 can be reduced to 3 and 4 (by dividing both numbers by their common factor, 4). A right triangle with the 2 perpendicular sides given in the ratio of 3:4 will always have a hypotenuse of 5. This suggests that the right triangle is a special 3-4-5 or 3:4:5 triangle.

The hypotenuse, then, is: \(\frac{\mathrm{h} }{\mathrm{12}} =\frac{\mathrm{5} }{\mathrm{3}}\)

\(h=12(\frac{\mathrm{5} }{\mathrm{3}})=\frac{\mathrm{60} }{\mathrm{3}}=20\)

The result will be the same using \(\frac{\mathrm{h} }{\mathrm{16}} =\frac{\mathrm{5} }{\mathrm{4}}\)

\(h=16(\frac{\mathrm{5} }{\mathrm{4}})=\frac{\mathrm{80} }{\mathrm{4}}=20 \)

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