Mary thinks the triangle is equilateral. How would you support or dispute her conjecture using trigonometric functions on right triangles?

• A. Dispute her conjecture, because in a right triangle, one angle is always $90°$, so the triangle cannot be equilateral.
• B. Support her conjecture, because all triangles with a $90°$ angle are equilateral.
• C. Dispute her conjecture, because the hypotenuse in a right triangle is always shorter than the other sides.
• D. Support her conjecture, because the sine and cosine of all angles in a right triangle are equal.

Answer: (A)

Verified step by step guidance

1. Recall the definition of an equilateral triangle: all three sides are equal in length, and all three interior angles are equal to 60°.
2. Recall the definition of a right triangle: one of its angles is exactly 90°, which is a key property of right triangles.
3. Use the fact that the sum of interior angles in any triangle is 180°. Since a right triangle has one 90° angle, the other two angles must sum to 90°, so they cannot both be 60°.
4. Apply trigonometric functions such as sine and cosine to the angles of the right triangle to verify their measures. For example, if one angle is 90°, the sine and cosine values of the other angles will not both correspond to 60° angles.
5. Conclude that because a right triangle must have a 90° angle and the other two angles cannot both be 60°, it cannot be equilateral, thus disputing Mary’s conjecture.