Question

in circle o, $overline{rt}$ and $overline{su}$ are diameters. if $mwidehat{rv}=mwidehat{vu}$, what is $mwidehat{vu}$?
$47^{circ}$
$52^{circ}$
$64^{circ}$
$87^{circ}$

Explanation:

Step1: Set up an equation

Since $\overline{RT}$ is a diameter, the sum of the central - angles $\angle RO V$ and $\angle VO U$ and $\angle UOT$ is $180^{\circ}$. Given that $\angle RO V=(13x)^{\circ}$ and $\angle UOT=(15x - 8)^{\circ}$, and $m\overline{RV}=m\overline{VU}$, so $\angle RO V=\angle VO U=(13x)^{\circ}$. Then the equation is $13x+13x+(15x - 8)=180$.

Step2: Simplify the equation

Combine like - terms: $(13x + 13x+15x)-8 = 180$, which gives $41x-8 = 180$.

Step3: Solve for $x$

Add 8 to both sides of the equation: $41x=180 + 8=188$. Then divide both sides by 41: $x = 4.6$. But we can also use the fact that vertical angles are equal. $\angle RO V$ and $\angle UOT$ are vertical angles, so $13x=15x - 8$.
Subtract $13x$ from both sides: $0=15x-13x - 8$, which simplifies to $2x-8 = 0$. Add 8 to both sides: $2x=8$, and then $x = 4$.

Step4: Find $m\overline{VU}$

Since $m\overline{RV}=m\overline{VU}$ and $\angle RO V=\angle VO U=(13x)^{\circ}$, substitute $x = 4$ into $(13x)^{\circ}$. So $m\overline{VU}=13\times4=52^{\circ}$.

Answer:

$52^{\circ}$