Question

given the lengths 10\, 15\, 17\, classify the triangle formed and or state that no triangle can be formed. acute triangle right triangle no triangle obtuse triangle

Explanation:

Step1: Check triangle inequality

For three side lengths \(a = 10\), \(b=15\), \(c = 17\) (where \(c\) is the longest side), check \(a + b>c\).
\(10+15=25>17\), \(10 + 17=27>15\), \(15+17 = 32>10\). So a triangle can be formed.

Step2: Use the Pythagorean - like inequality

For a triangle with side lengths \(a\), \(b\), \(c\) (\(c\) is the longest side), if \(a^{2}+b^{2}>c^{2}\), the triangle is acute; if \(a^{2}+b^{2}=c^{2}\), the triangle is right; if \(a^{2}+b^{2}Calculate \(a^{2}+b^{2}\) and \(c^{2}\):
\(a^{2}+b^{2}=10^{2}+15^{2}=100 + 225=325\)
\(c^{2}=17^{2}=289\)

Since \(a^{2}+b^{2}=325>c^{2}=289\)

Answer:

Acute triangle