Question

given a right triangle with leg lengths as shown, what is the length of the hypotenuse? image of right triangle with legs √45 and √27 options: 72, √2754, 6√2, √15

Explanation:

Step1: Recall Pythagorean theorem

For a right triangle, \( c = \sqrt{a^2 + b^2} \), where \( a, b \) are legs, \( c \) is hypotenuse. Here, \( a = \sqrt{45} \), \( b = \sqrt{27} \).

Step2: Substitute into the formula

\( c = \sqrt{(\sqrt{45})^2 + (\sqrt{27})^2} \)

Step3: Simplify the squares

\( (\sqrt{45})^2 = 45 \), \( (\sqrt{27})^2 = 27 \). So \( c = \sqrt{45 + 27} \)

Step4: Add the numbers

\( 45 + 27 = 72 \), so \( c = \sqrt{72} \)

Step5: Simplify the square root

\( \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \)

Answer:

\( 6\sqrt{2} \) (corresponding to the option "6√2")