Question

1. if $overrightarrow{bd}perpoverrightarrow{ac}$, $mangle dbe=(2x - 1)^{circ}$, and $mangle cbe=(5x - 42)^{circ}$, find the value of $x$.

Explanation:

Step1: Identify angle - relationship

Since $\overrightarrow{BD}\perp\overrightarrow{AC}$, $\angle DBC = 90^{\circ}$. And $\angle CBE+\angle DBE=\angle DBC$.

Step2: Set up the equation

We have $(2x - 1)+(5x - 42)=90$.

Step3: Simplify the left - hand side

Combine like terms: $2x+5x-1 - 42=90$, which gives $7x-43 = 90$.

Step4: Solve for x

Add 43 to both sides: $7x=90 + 43$, so $7x=133$. Then divide both sides by 7: $x=\frac{133}{7}=19$.

Answer:

$x = 19$