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3GTG 6

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A triangle has sides of integer lengths 3, 6 and x. For how many values of x will the triangle be acute?

Dec 9, 2018

#1
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$$\text{The triangle will be acute for }\\ 3 < x < \sqrt{3^2 + 6^2} = \\ 3 < x < 3\sqrt{5}\\ \text{That's a continuum of values so asking how many there are is sort of meaningless}$$

Dec 9, 2018
edited by Rom  Dec 9, 2018
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Actually, the answer key says it is 1. Do you know how?

dgfgrafgdfge111  Dec 9, 2018
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oh it says integer lengths... DOH

$$3 < x < 3 \sqrt{5} \approx 6.7\\ \text{That gives you 4, 5, and 6}$$

4 gives you an angle of about 36.36 degrees

5 gives you an angle of about 56.25 degrees

6 gives you an angle of about 75.53 degrees

all 3 are acute triangles

Rom  Dec 10, 2018