Question

in the parallelogram below, if $overline{ab}=9y + 20$ and $overline{dc}=28 + 5y$. find the following. $y = $ $overline{ab}=$ in the image below, assume that lines which appear tangent are tangent. given that $moverparen{cah}=210^{circ}$ and $moverparen{ch}=150^{circ}$, find the $mangle x$. select one: a. $30^{circ}$ b. $210^{circ}$ c. $60^{circ}$ d. $150^{circ}$

Explanation:

Step1: Use parallelogram property

In a parallelogram, opposite - sides are equal. So, $AB = DC$.
Set up the equation $9y + 20=28 + 5y$.

Step2: Solve the equation for y

Subtract $5y$ from both sides: $9y-5y + 20=28+5y - 5y$, which simplifies to $4y+20 = 28$.
Then subtract 20 from both sides: $4y+20 - 20=28 - 20$, giving $4y=8$.
Divide both sides by 4: $y=\frac{8}{4}=2$.

Step3: Find the length of AB

Substitute $y = 2$ into the expression for $AB$: $AB=9y + 20=9\times2+20=18 + 20=38$.

For the circle - related part:
The measure of an angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs.
The formula is $m\angle x=\frac{1}{2}(m\overparen{CAH}-m\overparen{CH})$.
Given $m\overparen{CAH}=210^{\circ}$ and $m\overparen{CH}=150^{\circ}$.

Step4: Calculate m∠x

$m\angle x=\frac{1}{2}(210 - 150)=\frac{1}{2}\times60 = 30^{\circ}$.

Answer:

$y = 2$, $AB = 38$, A. $30^{\circ}$