Question

which statement about △abc is true? o $overline{ac}$ is shorter than $overline{ab}$. o $overline{ab}$ is longer than $overline{bc}$. o $overline{ac}$ is the longest side of △abc. o $overline{bc}$ is the shortest side of △abc.

Explanation:

Step1: Recall angle - side relationship

In a triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.

Step2: Identify largest and smallest angles

In $\triangle ABC$, $\angle A = 61^{\circ}$, $\angle B=59^{\circ}$, $\angle C = 60^{\circ}$. The largest angle is $\angle A$ and the smallest angle is $\angle B$.

Step3: Determine opposite sides

The side opposite $\angle A$ is $BC$, the side opposite $\angle B$ is $AC$, and the side opposite $\angle C$ is $AB$. Since $\angle B$ is the smallest angle, the side opposite it, $AC$, is the shortest side. Since $\angle A$ is the largest angle, the side opposite it, $BC$, is the longest side.

• Analyze option 1: $AC$ is shorter than $AB$. Since $\angle C>\angle B$, $AB > AC$, this is true.
• Analyze option 2: $AB$ is longer than $BC$. Since $\angle C<\angle A$, $BC>AB$, this is false.
• Analyze option 3: $AC$ is the longest side. Since $\angle B$ is the smallest angle, $AC$ is the shortest side, this is false.
• Analyze option 4: $BC$ is the shortest side. Since $\angle A$ is the largest angle, $BC$ is the longest side, this is false.

Answer:

$\overline{AC}$ is shorter than $\overline{AB}$.