Answer: [tex]\boxed{\{24, 45, 51\}}[/tex]
Step-by-step explanation:
Hi there! In order to see if the sides could represent a right triangle we can use the Pythagorean theorem, which states that in a right triangle the square of the lengths of the two smaller sides is equal to the hypotenuse(the largest side opposite to the right angle). In order to check, we must take the two smaller sides, square them and see if they equal to the square of the larger side.
Pythagorean theorem: [tex]\[ \boxed{a^2 + b^2 = c^2} \][/tex]
Solving:
[tex]\begin{document}\textbf{For $\{42, 55, 70\}$:}\[\begin{aligned}42^2 + 55^2 = 1764 + 3025 \\= 4789 \\70^2 = 4900\end{aligned}\]These numbers do not satisfy the Pythagorean theorem. \end{document}[/tex] ❌
[tex]\rule{\linewidth}{0.5mm}[/tex][tex]\documentclass{}\begin{document}\textbf{For $\{24, 45, 51\}$:}\[\begin{aligned}24^2 + 45^2 = 576 + 2025 \\= 2601 \\51^2 = 2601\end{aligned}\]These numbers satisfy the Pythagorean theorem. \ding{}\\\end{document}[/tex] ✅
[tex]\rule{\linewidth}{0.5mm}[/tex]
[tex]\documentclass{}\usepackage{}\usepackage{\begin{document}\\\textbf{For $\{30, 39, 50\}$:}\\\[\begin{aligned}30^2 + 39^2 = 900 + 1521 \\= 2421 \\50^2 = 2500\end{aligned}\]These numbers do not satisfy the Pythagorean theorem. \\\end{document}[/tex] ❌
[tex]\rule{\linewidth}{0.5mm}[/tex]
[tex]\documentclass{}\usepackage{}\usepackage{}\begin{document}\textbf{For $\{28, 45, 54\}$:}\[\begin{aligned}28^2 + 45^2 = 784 + 2025 \\= 2809 \\54^2 = 2916\end{aligned}\]These numbers do not satisfy the Pythagorean theorem. \\\end{document}[/tex]❌
[tex]\rule{\linewidth}{0.5mm}[/tex]
That's it!
