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)

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

90°
90°

74°
74°

53°
53°

106°

GPT-4o mini · Answer

To find the arc measure of $ MN $, we can use the properties of inscribed angles. Since triangle $ LMN $ is inscribed in the circle and has diameter $ LM $, we know that angle $ N $ will be a right angle (90°) because any angle inscribed in a semicircle is a right angle. Using the information given: 1. Measure of angle $ L = 37° $. 2. Therefore, measure of angle $ N $ is $ 90° $. The sum of the angles in triangle $ LMN $ is $ 180° $: $$ \text{angle } L + \text{angle } M + \text{angle } N = 180° $$ Substituting the known angles: $$ 37° + \text{angle } M + 90° = 180° $$ $$ \text{angle } M = 180° - 127° = 53° $$ Now, to find the arc $ MN $. The measure of arc $ MN $ is given by twice the measure of the inscribed angle $ M $ that subtends it: $$ \text{Arc } MN = 2 \times \text{angle } M = 2 \times 53° = 106°. $$ Therefore, the arc measure of $ MN $ is $ \boxed{106°} $.