Question

which list orders the sides of triangle abc from longest to shortest length?

ac, bc, ab
ab, bc, ac
bc, ac, ab
ac, ab, bc

Explanation:

Find the missing angle

In triangle \(ABC\), the sum of angles is \(180^\circ\).
Given:

• \(\angle C = 90^\circ\) (right angle)
• \(\angle B = 30^\circ\)

Calculate \(\angle A\):

$$ \angle A = 180^\circ - 90^\circ - 30^\circ = 60^\circ $$

Compare the angle measures

Order the angles from largest to smallest:

$$ \angle C (90^\circ) > \angle A (60^\circ) > \angle B (30^\circ) $$

Relate angles to opposite sides

By the Angle-Side Relationships in a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

• Opposite \(\angle C (90^\circ)\) is side \(AB\).
• Opposite \(\angle A (60^\circ)\) is side \(BC\).
• Opposite \(\angle B (30^\circ)\) is side \(AC\).

Therefore, the sides ordered from longest to shortest are:

$$ AB > BC > AC $$

Answer:

• AC, BC, AB
• AB, BC, AC (Correct answer)
• BC, AC, AB
• AC, AB, BC