Question

1. what is the perimeter of $\triangle qts$?

Explanation:

Step1: Identify triangle properties

In triangle \( QTS \), \( TQ = TS = 6 \), so it's isosceles with \( TQ = TS \). The altitude from \( T \) to \( QS \) bisects \( QS \), so \( 3x - 1 = 4 - 2x \).

Step2: Solve for \( x \)

\[

$$\begin{align*} 3x - 1 &= 4 - 2x \\ 3x + 2x &= 4 + 1 \\ 5x &= 5 \\ x &= 1 \end{align*}$$

\]

Step3: Find lengths of \( QS \) segments

Substitute \( x = 1 \) into \( 3x - 1 \): \( 3(1) - 1 = 2 \).
Substitute \( x = 1 \) into \( 4 - 2x \): \( 4 - 2(1) = 2 \).
Thus, \( QS = 2 + 2 = 4 \).

Step4: Calculate perimeter

Perimeter of \( \triangle QTS = TQ + TS + QS = 6 + 6 + 4 = 16 \).

Answer:

The perimeter of \( \triangle QTS \) is \( \boldsymbol{16} \).