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Geometry Level 1

Given 3, 4, and 5 are valid unit lengths for the sides of a right triangle, which of the answer options does not give valid unit lengths for the sides of a right triangle?

27, 36, 45

300, 400, 500

\frac{1}{3}, \frac{1}{4}, \frac{1}{5}

\frac{3}{2}, \frac{4}{2}, \frac{5}{2}

1 solution

Jason Dyer Staff
Oct 21, 2016

Three of the trios take the numbers 3, 4, and 5 and multiply each by the same factor. Thus the triangles are similar to the 3-4-5 right triangle and are themselves right triangles.

$3 \times 100 = 300, 4 \times 100 = 400, 5 \times 100 = 500$

$3 \times \frac{1}{2} = \frac{3}{2}, 4 \times \frac{1}{2} = \frac{4}{2}, 5 \times \frac{1}{2} = \frac{5}{2}$

$3 \times 9 = 27, 4 \times 9 = 36, 5 \times 9 = 45$

It can be verified the trio $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ doesn't work by testing with the Pythagorean Theorem: $\sqrt{\left(\frac{1}{3}\right)^2 + \left(\frac{1}{4}\right)^2} = \frac{5}{12} ,$ not $\frac{1}{5} .$

In general, we can show that if $(a,b,c)$ is a pythagorean triplet, then $(\frac{1}{a} , \frac{1}{b}, \frac{c}{ab})$ would be the the pythagorean triplet that we're looking for.

Calvin Lin Staff - 4 years, 7 months ago

@Jason Dyer I think you should say (1/4)^2 + (1/5)^2 not equal to (1/3)^2 because 1/3 is the largest.

Dexter Woo Teng Koon - 4 years, 7 months ago

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