Question

the numbers 7, 24, 25 form a pythagorean triple. which of these sets are the side lengths of triangles similar to a triangle whose side lengths measure 7, 24, 25? select all the correct answers. 14, 48, 50 9, 12, 15 2, √20, 2√6 8, 15, 17 √7, √24, √25 35, 120, 125 21, 72, 78

Explanation:

Step1: Recall similarity condition

Triangles are similar if their side lengths are in proportion (i.e., each side of one triangle is a scalar multiple of the corresponding side of the other triangle). For the triple (7,24,25), we check if each option can be written as \(k(7,24,25)\) for some positive real number \(k\).

Step2: Check 14,48,50

Divide each term by 2: \(\frac{14}{2}=7\), \(\frac{48}{2}=24\), \(\frac{50}{2}=25\). This is \(2\times(7,24,25)\).
<Expression>
\(14=2\times7\), \(48=2\times24\), \(50=2\times25\)
</Expression>

Step3: Check 9,12,15

This is \(3\times(3,4,5)\), which is a different Pythagorean triple, not a multiple of (7,24,25).
<Expression>
\(9=3\times3\), \(12=3\times4\), \(15=3\times5\)
</Expression>

Step4: Check \(2,\sqrt{20},2\sqrt{6}\)

Simplify \(\sqrt{20}=2\sqrt{5}\), \(2\sqrt{6}\approx4.899\). No positive \(k\) makes \(k\times7=2\), \(k\times24=2\sqrt{5}\) (since \(\frac{2}{7}
eq\frac{2\sqrt{5}}{24}\)).

Step5: Check 8,15,17

This is a distinct Pythagorean triple, not a multiple of (7,24,25).

Step6: Check \(\sqrt{7},\sqrt{24},\sqrt{25}\)

\(\sqrt{25}=5\), and \(\frac{\sqrt{7}}{7}=\frac{1}{\sqrt{7}}\), \(\frac{\sqrt{24}}{24}=\frac{1}{\sqrt{24}}\), which are not equal, so no common scalar \(k\).

Step7: Check 35,120,125

Divide each term by 5: \(\frac{35}{5}=7\), \(\frac{120}{5}=24\), \(\frac{125}{5}=25\). This is \(5\times(7,24,25)\).
<Expression>
\(35=5\times7\), \(120=5\times24\), \(125=5\times25\)
</Expression>

Step8: Check 21,72,78

Divide 21 by 3=7, 72 by 3=24, 78 by 3=26≠25. Not a multiple of (7,24,25).
<Expression>
\(21=3\times7\), \(72=3\times24\), \(78=3\times26\)
</Expression>

Answer:

14, 48, 50
35, 120, 125