Which of the following triangles is both scalene and acute?

• A. A triangle with side lengths $5$, $6$, and $7$
• B. A triangle with angles $90°$, $45°$, and $45°$
• C. A triangle with side lengths $3$, $4$, and $5$
• D. A triangle with side lengths $2$, $2$, and $3$

Answer: (A)

Verified step by step guidance

1. Recall the definitions: a scalene triangle has all sides of different lengths, and an acute triangle has all interior angles less than 90°.
2. Check each triangle to see if it is scalene by comparing the side lengths. For example, the triangle with sides 5, 6, and 7 has all different side lengths, so it is scalene.
3. Determine if each triangle is acute by examining its angles or using the side lengths with the Law of Cosines. For the triangle with sides 5, 6, and 7, use the Law of Cosines to find each angle and verify that all are less than 90°.
4. For the triangle with angles 90°, 45°, and 45°, note that it has a right angle, so it is not acute.
5. For the triangle with sides 3, 4, and 5, recognize it as a right triangle (Pythagorean triple), so it is not acute. For the triangle with sides 2, 2, and 3, check if it is scalene (it is isosceles, since two sides are equal), so it is not scalene.