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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

A right triangle has legs with lengths of $24$ centimeters and $21$ centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form $3 StartRoot d EndRoot$ , where $d$ is an integer, what is the value of $d$ ?

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Explanation

The correct answer is $113$ . It's given that the legs of a right triangle have lengths $24$ centimeters and $21$ 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 $h$ represents the length, in centimeters, of the hypotenuse of the right triangle, $h^{2} = 24^{2} + 21^{2}$ . This equation is equivalent to $h squared equals 1,017$ . Taking the square root of each side of this equation yields $h equals StartRoot 1,017 EndRoot$ . This equation can be rewritten as $h equals StartRoot 9 dot 113 EndRoot$ , or $h equals StartRoot 9 EndRoot dot StartRoot 113 EndRoot$ . This equation is equivalent to $h equals 3 StartRoot 113 EndRoot$ . It's given that the length of the triangle's hypotenuse, in centimeters, can be written in the form $3 StartRoot d EndRoot$ . It follows that the value of $d$ is $113$ .