Question

find the longest side of $\triangle stu$.

Explanation:

Step1: Set up angle sum equation

The sum of angles in a triangle is $180^\circ$, so:
$$(7a-78) + 64 + (a+58) = 180$$

Step2: Simplify and solve for $a$

Combine like terms and isolate $a$:
$$8a + 44 = 180$$
$$8a = 180 - 44$$
$$8a = 136$$
$$a = \frac{136}{8} = 17$$

Step3: Calculate each angle

Substitute $a=17$ into each angle expression:

• $\angle U = 7(17)-78 = 119-78 = 41^\circ$
• $\angle S = 17+58 = 75^\circ$
• $\angle T = 64^\circ$

Step4: Match largest angle to opposite side

The largest angle is $\angle S = 75^\circ$, and its opposite side is $TU$.

Answer:

The longest side of $\Delta STU$ is $\boldsymbol{TU}$.