Question

the sides of a triangle have lengths 6, 15, and 15. what kind of triangle is it?
acute
right
obtuse

Explanation:

Step1: Identify the longest side

The sides are 6, 15, 15. The longest side length is 15.

Step2: Check triangle type using the law of cosines (for the angle opposite the longest side, but since two sides are equal, it's isoceles. Also, check the angle: for a triangle with sides \(a = 6\), \(b = 15\), \(c = 15\), we can check the angle opposite \(a\) (the smallest side). Using the law of cosines: \(\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\). Plugging in: \(\cos A=\frac{15^{2}+15^{2}-6^{2}}{2\times15\times15}=\frac{225 + 225- 36}{450}=\frac{414}{450}=0.92\). Since \(\cos A>0\), angle \(A\) is acute. Also, since two sides are equal (15 and 15), it's an isoceles triangle, and the angles opposite equal sides are equal. The largest angles are opposite the equal sides (15), and since the angle opposite 6 is acute, the other angles (which are equal) are also acute (because in a triangle, if the smallest angle is acute, and two sides are equal, the other angles are equal and since the sum of angles is \(180^{\circ}\), they must be acute as well. Alternatively, for a triangle with two equal sides (isoceles) and the third side shorter than the equal sides, and the angle opposite the shorter side is acute, so the triangle is acute. Wait, but also, let's check the other way: for a triangle with sides \(a,b,c\) ( \(c\) is longest), if \(a^{2}+b^{2}>c^{2}\), it's acute; \(a^{2}+b^{2}=c^{2}\) right; \(a^{2}+b^{2}0\), angle \(C\) is acute. Then angles \(A\) and \(B\) are equal (since \(a = b\)), so \(A = B=\frac{180 - C}{2}\). Since \(C\) is acute (\(C<90^{\circ}\)), then \(180 - C>90^{\circ}\), so \(A = B>\frac{90^{\circ}}{2}=45^{\circ}\), but are they acute? Yes, because \(A + B + C=180\), so \(A = B=\frac{180 - C}{2}\). If \(C<90\), then \(180 - C>90\), so \(A = B>45\), but since \(C\) is, say, \(C=\arccos(0.92)\approx23.07^{\circ}\), then \(A = B=\frac{180 - 23.07}{2}\approx78.46^{\circ}\), which is acute. So the triangle is acute.

Answer:

acute (the triangle with sides 6,15,15 is an acute isoceles triangle)