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Right triangles and trigonometry SAT question

Math · Geometry and Trigonometry · Hard · Answer it here, explanation included.

A right triangle has legs with lengths of centimeters and centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form , where is an integer, what is the value of ?

Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.

ExplanationShow

The correct answer is . It's given that the legs of a right triangle have lengths centimeters and centimeters. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. It follows that if represents the length, in centimeters, of the hypotenuse of the right triangle, . This equation is equivalent to . Taking the square root of each side of this equation yields . This equation can be rewritten as , or . This equation is equivalent to . It's given that the length of the triangle's hypotenuse, in centimeters, can be written in the form . It follows that the value of is .

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