Multiple Choice
Which of the following correctly states the Law of Sines for triangle with sides , , opposite angles , , ?
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7. Non-Right Triangles
Law of Sines
Multiple Choice
Given that the major arc of a circle measures , which of the following best describes triangle inscribed in the circle with points , , and on the circumference?
A
Triangle is an obtuse triangle because the inscribed angle opposite the major arc is less than .
B
Triangle is an obtuse triangle because the inscribed angle opposite the major arc is greater than .
C
Triangle is an acute triangle because all its angles are less than .
D
Triangle is a right triangle because the inscribed angle opposite the major arc is exactly .
Verified step by step guidance1
Recall that the measure of an inscribed angle in a circle is half the measure of the intercepted arc. This means if an inscribed angle intercepts an arc of measure \( \theta \), then the angle measure is \( \frac{\theta}{2} \).
Identify the arc opposite the inscribed angle at point \( M \). Since the major arc \( JL \) measures \( 300^\circ \), the minor arc \( JL \) (the other arc between points \( J \) and \( L \)) measures \( 360^\circ - 300^\circ = 60^\circ \).
Determine which arc the inscribed angle at \( M \) intercepts. The angle at \( M \) intercepts the major arc \( JL \) of \( 300^\circ \), so the measure of angle \( M \) is half of \( 300^\circ \).
Calculate the measure of angle \( M \) using the inscribed angle theorem: \( \text{angle } M = \frac{300^\circ}{2} \). This will give an angle greater than \( 90^\circ \), indicating an obtuse angle.
Since one angle of triangle \( JLM \) is obtuse (greater than \( 90^\circ \)), conclude that triangle \( JLM \) is an obtuse triangle.
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