Question

determine if each set of numbers can be the lengths of the sides of a right triangle. select the correct text in the table.

Explanation:

Step1: Check the first set (5,12,13)

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\).
\(5^{2}+12^{2}=25 + 144=169\), and \(13^{2}=169\). So \(5^{2}+12^{2}=13^{2}\).

Step2: Check the second set \((12,35,20\sqrt{3})\)

\(12^{2}+35^{2}=144+1225 = 1369\), \((20\sqrt{3})^{2}=400\times3=1200\). Since \(1369
eq1200\).

Step3: Check the third set \((5,10,5\sqrt{5})\)

\(5^{2}+10^{2}=25 + 100=125\), \((5\sqrt{5})^{2}=25\times5 = 125\). So \(5^{2}+10^{2}=(5\sqrt{5})^{2}\).

Step4: Check the fourth set \((8,12,15)\)

\(8^{2}+12^{2}=64+144 = 208\), \(15^{2}=225\). Since \(208
eq225\).

Step5: Check the fifth set \((20,99,101)\)

\(20^{2}+99^{2}=400+9801=10201\), \(101^{2}=10201\). So \(20^{2}+99^{2}=101^{2}\).

Answer:

\(a\)	\(b\)	\(c\)	Pythagorean triple?
\(12\)	\(35\)	\(20\sqrt{3}\)	No
\(5\)	\(10\)	\(5\sqrt{5}\)	Yes
\(8\)	\(12\)	\(15\)	No
\(20\)	\(99\)	\(101\)	Yes