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Find Measure of Angle M in Right Triangle with Legs 7 and 15
Mathematics
Grade 10 (High School Geometry)
Question Content
Use the diagram (right triangle MPN with right angle at P, PN=15, MP=7) to find $m\\angle M$. Round your answer to the nearest tenth.
Correct Answer
65.0°
Detailed Solution Steps
1
Step 1: Identify the sides relative to $\\angle M$: opposite side = PN=15, adjacent side = MP=7.
2
Step 2: Use the tangent function, which relates opposite and adjacent sides: $\\tan(\\angle M) = \\frac{opposite}{adjacent} = \\frac{15}{7}$.
3
Step 3: Calculate the inverse tangent to find the angle: $\\angle M = \\tan^{-1}(\\frac{15}{7}) \\approx \\tan^{-1}(2.1429) \\approx 65.0°$.
Knowledge Points Involved
1
Tangent Function for Right Triangles
For an acute angle in a right triangle, the tangent of the angle is the ratio of the length of the opposite side to the length of the adjacent side: $\\tan(\\theta) = \\frac{opposite}{adjacent}$. The inverse tangent function ($\\tan^{-1}$) is used to find the measure of an angle when the ratio of opposite to adjacent sides is known.
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