in the diagram of circle a, what is m∠lmn? 75° 90° 120° 135°
in the diagram of circle a, what is m∠lmn? 75° 90° 120° 135°
Answer
Explanation:
Step1: Recall the formula for the measure of an angle formed by two tangents
The measure of an angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs. Let the major - arc be $LPN = 270^{\circ}$ and the minor - arc be $LN$. The sum of the measures of the major and minor arcs of a circle is $360^{\circ}$. So, the measure of arc $LN=360 - 270=90^{\circ}$. The formula for $\angle LMN$ (where $LM$ and $MN$ are tangents) is $\angle LMN=\frac{1}{2}(m\overset{\frown}{LPN}-m\overset{\frown}{LN})$.
Step2: Substitute the arc - measures into the formula
Substitute $m\overset{\frown}{LPN} = 270^{\circ}$ and $m\overset{\frown}{LN}=90^{\circ}$ into the formula: $\angle LMN=\frac{1}{2}(270 - 90)$. First, calculate the value inside the parentheses: $270−90 = 180$. Then, $\frac{1}{2}\times180 = 90^{\circ}$.
Answer:
$90^{\circ}$