Question

hd is the perpendicular bisector of ky such that point d lies on ky. ky = 7y - 14, hy = 4y + 23, and hk = 6y + 7. when the perpendicular - bisector theorem is applied, what is the value of y? y = units. ∠abc is shown below. (overline{bd}) is the angle bisector of ∠abc. m∠abc = 64°. what is the measure of ∠dbc? °

Explanation:

Step1: Apply perpendicular - bisector theorem

Since $HD$ is the perpendicular bisector of $KY$, by the perpendicular - bisector theorem, $HK = HY$. So we set up the equation $6y + 7=4y + 33$.

Step2: Solve the equation for $y$

Subtract $4y$ from both sides: $6y-4y + 7=4y-4y + 33$, which simplifies to $2y+7 = 33$. Then subtract 7 from both sides: $2y+7 - 7=33 - 7$, getting $2y=26$. Divide both sides by 2: $\frac{2y}{2}=\frac{26}{2}$, so $y = 13$.

Step3: Find measure of $\angle DBC$

Since $\overline{BD}$ is the angle - bisector of $\angle ABC$ and $m\angle ABC = 64^{\circ}$, by the definition of an angle - bisector, $m\angle DBC=\frac{1}{2}m\angle ABC$. So $m\angle DBC=\frac{1}{2}\times64^{\circ}=32^{\circ}$.

Answer:

$y = 13$
$m\angle DBC = 32^{\circ}$