Question

select the correct answer.
a building has a triangular rooftop terrace which is modeled by triangle abc. in triangle abc, the measure of ∠a is 55°, the measure of ∠b is 65°, and the measure of ∠c is 60°.
which side of the terrace has the greatest length?
a. (overline{ab})
b. (overline{bc})
c. cannot be determined
d. (overline{ac})

Explanation:

Step1: Recall Triangle Angle-Side Relationship

In a triangle, the longest side is opposite the largest angle (Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$, so larger angle $\implies$ longer opposite side).

Step2: Calculate $\angle B$ (Wait, angles given: $\angle A = 55^\circ$, $\angle B = 65^\circ$, $\angle C = 60^\circ$). Find the largest angle.

Largest angle: $\angle B = 65^\circ$ (since $65^\circ > 60^\circ > 55^\circ$).

Step3: Identify Side Opposite $\angle B$

Side opposite $\angle B$ is $\overline{AC}$? Wait no: In $\triangle ABC$, side opposite $\angle A$ is $\overline{BC}$, opposite $\angle B$ is $\overline{AC}$, opposite $\angle C$ is $\overline{AB}$. Wait, no: Wait, vertex labels: $\angle A$ is at vertex $A$, so side opposite $\angle A$ is $BC$ (side $a$), opposite $\angle B$ is $AC$ (side $b$), opposite $\angle C$ is $AB$ (side $c$). Wait, but $\angle B = 65^\circ$ is the largest angle. So side opposite $\angle B$ is $AC$? Wait no, wait the angles: $\angle A = 55$, $\angle B = 65$, $\angle C = 60$. So largest angle is $\angle B = 65^\circ$. So side opposite $\angle B$ is $AC$? Wait no, wait: In triangle $ABC$, angle at $A$: $\angle A$, side opposite is $BC$ (length $a$). Angle at $B$: $\angle B$, side opposite is $AC$ (length $b$). Angle at $C$: $\angle C$, side opposite is $AB$ (length $c$). Wait, but the question is which side has greatest length. So largest angle is $\angle B = 65^\circ$, so side opposite $\angle B$ is $AC$? Wait no, wait the options: A. $\overline{AB}$, B. $\overline{BC}$, D. $\overline{AC}$. Wait, maybe I mixed up. Wait, let's re-express:

Angles:
$\angle A = 55^\circ$ (opposite side $BC$),
$\angle B = 65^\circ$ (opposite side $AC$),
$\angle C = 60^\circ$ (opposite side $AB$).

Largest angle: $\angle B = 65^\circ$, so side opposite is $AC$? Wait no, wait the options: D is $\overline{AC}$. Wait, but let's check again. Wait, maybe I made a mistake. Wait, angles: $55$, $65$, $60$. So $65$ is largest. Side opposite $65^\circ$ (angle $B$) is $AC$ (side $b$). So $AC$ is opposite the largest angle, so $AC$ is the longest side. Wait, but the options: D is $\overline{AC}$. Wait, but let's confirm.

Law of Sines: $\frac{BC}{\sin A} = \frac{AC}{\sin B} = \frac{AB}{\sin C}$. Since $\sin B$ is largest (because $B = 65^\circ$ is largest angle, and $\sin$ increases from $0^\circ$ to $90^\circ$), so $AC = \frac{BC \cdot \sin B}{\sin A}$, so larger $\sin B$ implies longer $AC$. So yes, $AC$ is longest. Wait, but the options: D is $\overline{AC}$. So the answer is D.

Answer:

D. $\overline{AC}$