Question

which value of x would make \\(\overline{tv} \parallel \overline{qs}\\)? 10 3 8 11

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' theorem)

If \( \overline{TV} \parallel \overline{QS} \), then by the Basic Proportionality Theorem, \( \frac{RT}{TQ} = \frac{RV}{VS} \).
Given \( RT = x + 4 \), \( TQ = x - 3 \), \( RV = x + 10 \), and \( VS = x \). So we set up the proportion:

$$ \frac{x + 4}{x - 3} = \frac{x + 10}{x} $$

Step2: Cross - multiply to solve the equation

Cross - multiplying gives \( x(x + 4)=(x - 3)(x + 10) \).
Expand both sides:
Left side: \( x(x + 4)=x^{2}+4x \)
Right side: \( (x - 3)(x + 10)=x^{2}+10x-3x - 30=x^{2}+7x - 30 \)
So the equation becomes \( x^{2}+4x=x^{2}+7x - 30 \).

Step3: Simplify the equation

Subtract \( x^{2} \) from both sides: \( 4x = 7x-30 \).
Subtract \( 4x \) from both sides: \( 0 = 3x - 30 \).
Add 30 to both sides: \( 3x=30 \).
Divide both sides by 3: \( x = 10 \).

Answer:

10