Question

11 mark for review a right triangle has legs with lengths of 24 centimeters and 21 centimeters. if the length of this triangles hypotenuse, in centimeters, can be written in the form 3√d, where d is an integer, what is the value of d?

Explanation:

Step1: Apply Pythagorean theorem

For a right triangle, \( c^2 = a^2 + b^2 \), where \( a = 24 \), \( b = 21 \), and \( c \) is the hypotenuse. So \( c^2 = 24^2 + 21^2 \).
Calculate \( 24^2 = 576 \) and \( 21^2 = 441 \). Then \( c^2 = 576 + 441 = 1017 \).

Step2: Express \( c \) in the form \( 3\sqrt{d} \)

We know \( c = \sqrt{1017} \). Factor 1017: \( 1017 = 9\times113 \), so \( \sqrt{1017} = \sqrt{9\times113} = 3\sqrt{113} \). Thus, \( d = 113 \).

Answer:

113