Question

find the shortest side of \\( \triangle def \\).

Explanation:

Step1: Find the value of \( t \)

Use the triangle - angle sum theorem (\( \angle E+\angle F+\angle D = 180^{\circ}\)).

$$19t+21t+(t + 57)=180$$

$$41t+57 = 180$$

$$41t=180 - 57$$

$$41t=123$$

$$t = 3$$

Step2: Calculate the angles

• \(\angle E=19t\), substitute \( t = 3 \), then \(\angle E=19\times3=57^{\circ}\)
• \(\angle F=21t\), substitute \( t = 3 \), then \(\angle F=21\times3 = 63^{\circ}\)
• \(\angle D=t + 57\), substitute \( t = 3 \), then \(\angle D=3+57=60^{\circ}\)

Step3: Use the angle - side relationship

In a triangle, the side opposite the smallest angle is the shortest side.
Since \(\angle E = 57^{\circ}\), \(\angle D=60^{\circ}\), \(\angle F = 63^{\circ}\), the smallest angle is \(\angle E\).
The side opposite \(\angle E\) is \(DF\)

Answer:

\(DF\)