Lucky mistake

Geometry Level 3

When a student wrote down this problem from the board, they wrote down a number incorrectly.

In $\triangle ABC$ , $\angle B = 60^{\circ}$ , $AB=616$ , $BC= \boxed{???}$ . Find the length of side $AC$ .

The mistake: they accidentally left a zero off the end of $BC$ .

Lucky for them, they otherwise worked the problem correctly, and despite the transcription error they still got the answer correct!

To the nearest whole number, what is the answer to this math problem?

The answer is 590.

1 solution

Jeremy Galvagni
Aug 12, 2018

We know that they wrote $BC=n$ instead of $BC=10n$ . By the law of cosines (simplifying since $\cos{60^{\circ}}=1/2$ ) we must get the same answer either way:

$AC^{2}=616^{2}+n^{2}-616\cdot n = 616^{2}+(10n)^{2}-616\cdot 10n$

$5544n = 99n^{2}$

$n = 5544/99 = 56$ where they should have written down $560$ . Either way,

$AC^{2}=616^{2}+56^{2}-616 \cdot 56 = 348096$

$AC = \sqrt{348096} \approx 589.9966102 \approx \boxed{590}$

Lucky mistake

The answer is 590.

1 solution

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