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Question 1102362: In ΔABC, m∠CAB = 30°, M is the midpoint of AB so that AB = 2CM. Find the angles of the triangle. Find AB if BC = 7 ft. What are the angles of the triangle?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39629)   (Show Source):
Answer by ikleyn(52873)  (Show Source):
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In ΔABC, m∠CAB = 30°, M is the midpoint of AB so that AB = 2CM. Find the angles of the triangle.
Find AB if BC = 7 ft. What are the angles of the triangle?
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You are given the triangle  ABC, angle CAB is 30°, M is the midpoint of AB and AB = 2*CM.
In other words, |MA| = |MB| = |MC|.
It means that the points A, B and C are equidistant from the point M.
It implies that the point M is the center of the circle, circumscribed around the triangle ABC.
Then AB is the diameter of this circle and the angle ACB is learning on the diameter AB.
It implies that the angle ACB is the right angle and the triangle ABC is a right-angled triangle.
Since the acute angle CAB of this triangle is 30°, the other acute angle is 90°-30° = 60°.
Thus the triangle ABC is (30° - 60° - 90°) triangle.
Since the leg BC opposite to 30° is 7 ft long, the hypotenuse AB is twice as long as BC, i.e. 14 ft.
Answer. The angles of the triangle are 30°, 60° and 90°.
The length of AB is 14 ft.
Solved.
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