Question

major arc jl measures 300°
which describes triangle jlm?
right
obtuse
scalene
equilateral

Explanation:

Step1: Find the measure of minor arc JL

The total measure of a circle is \(360^\circ\). The major arc JL is \(300^\circ\), so the minor arc JL is \(360^\circ - 300^\circ=60^\circ\).

Step2: Recall the central angle theorem

The central angle \(\angle JML\) subtended by arc JL is equal to the measure of arc JL, so \(\angle JML = 60^\circ\).

Step3: Analyze the triangle JLM

In a circle, the radii \(MJ\) and \(ML\) are equal (since \(MJ\) and \(ML\) are radii of the same circle), so triangle JLM is isosceles with \(MJ = ML\). Also, we found that the included angle \(\angle JML=60^\circ\). In a triangle, if two sides are equal and the included angle is \(60^\circ\), then the triangle is equilateral (because if two sides are equal and the included angle is \(60^\circ\), the third side must also be equal to the other two sides by the law of cosines: \(JL^{2}=MJ^{2}+ML^{2}-2\cdot MJ\cdot ML\cdot\cos(60^\circ)\). Since \(MJ = ML\), let \(MJ = ML = r\). Then \(JL^{2}=r^{2}+r^{2}-2\cdot r\cdot r\cdot\frac{1}{2}=r^{2}+r^{2}-r^{2}=r^{2}\), so \(JL = r\), hence \(MJ = ML = JL\)).

Answer:

equilateral