1) 8 is a relatively small proportion of 24, so the length of the unknown side of each right triangle will be only a little less than 24. The value of x is double this. The only answer choice that makes sense is ...
45.25
2) The geometric mean is the square root of the product.
=√(9*16) = 3*4 = 12
3) The value of x must be between 9 and about 1.4 times 9, so the only answer that makes sense is
11.40
If you work through the math of the Pythagorean theorem, this choice is confirmed. x = √(9²+7²) = √130 ≈ 11.401754...
4) As with problem 3, the answer must lie between 5 and about 1.4*5 = 7. The only choice that makes sense is
5.83
Once again, if you work through the math, x = √(5²+3²) = √34 ≈ 5.83095189...
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If you remember that the hypotenuse of an isosceles right triangle is √2 ≈ 1.4 times the length of one side, you can often bound the possible answers to right triangle problems without doing much work at all. If the ratio of side lengths is smaller than 1:1, the length of the hypotenuse gets closer to the length of the longest leg.
