Question

find t.
v to z is a line segment with an arrow. v to x is 34, x to z is 16t - 4. w to y is a line segment with an arrow, w to x is 17, x to y is 12t - 14. t = blank submit

Explanation:

Step1: Identify Similar Triangles

The lines \( VZ \) and \( WY \) are parallel (indicated by the arrows), so \( \triangle VXZ \sim \triangle WXY \) by the Basic Proportionality Theorem (Thales' theorem). Thus, the sides are proportional: \( \frac{VX}{WX} = \frac{XZ}{XY} \).

Step2: Substitute Known Values

We know \( VX = 34 \), \( WX = 17 \), \( XZ = 16t - 4 \), and \( XY = 12t - 14 \). Substituting into the proportion: \( \frac{34}{17} = \frac{16t - 4}{12t - 14} \).

Step3: Simplify the Proportion

Simplify \( \frac{34}{17} = 2 \), so the equation becomes \( 2 = \frac{16t - 4}{12t - 14} \).

Step4: Solve for \( t \)

Multiply both sides by \( 12t - 14 \): \( 2(12t - 14) = 16t - 4 \).
Expand left side: \( 24t - 28 = 16t - 4 \).
Subtract \( 16t \) from both sides: \( 8t - 28 = -4 \).
Add 28 to both sides: \( 8t = 24 \).
Divide by 8: \( t = 3 \).

Answer:

\( t = 3 \)