Triangle LMN is inscribed inside a circle with diameter LM ; the inscribed angle is N . The angle measure at the vertex L is 37°. Using what you understand about inscribed angles, find the arc measure of MN .(1 point) Responses 106°

Question

Triangle LMN  is inscribed inside a circle with diameter LM ; the inscribed angle is N . The angle measure at the vertex L is 37°. Using what you understand about inscribed angles, find the arc measure of MN .(1 point) Responses 106°

GPT-4o mini · Answer

To find the arc measure of $ MN $ when triangle $ LMN $ is inscribed in a circle with diameter $ LM $ and the angle at vertex $ L $ measures $ 37^\circ $, we can use the properties of inscribed angles. The inscribed angle $ \angle L $ intercepts the arc $ MN $. According to the properties of inscribed angles, the measure of an inscribed angle is half the measure of the arc it intercepts. Let $ m $ be the measure of arc $ MN $. According to the relationship: $$ \angle L = \frac{1}{2} m $$ Given that $ \angle L = 37^\circ $, we can set up the equation: $$ 37^\circ = \frac{1}{2} m $$ To find $ m $, we multiply both sides by 2: $$ m = 2 \times 37^\circ = 74^\circ $$ Thus, the measure of the arc $ MN $ is $ 74^\circ $. There may be a misunderstanding with the response of $ 106^\circ $, as $ 74^\circ $ is the correct measure for arc $ MN $. If you're looking for the measure of the remaining arc (which may be referred to if a larger angle is being considered), the total circumference of a circle is $ 360^\circ $. Therefore, if the full circle is $ 360^\circ $: $$ \text{Measure of the remaining arc} = 360^\circ - 74^\circ = 286^\circ $$ But as per the original question focusing on arc $ MN $, the arc measure of $ MN $ is $ 74^\circ $.