In ΔABC, m∠CAB = 30°, M is the midpoint of AB so that AB = 2·MC. Find the angles of the triangle. Find AB if BC = 7 ft. (preferably with a picture)

Answer:The angles of the ΔABC are:[tex]m\angle A=30, m\angle B=60, m\angle C=90[/tex]AB= 14 ftStep-by-step explanation:Given:A triangle ABC, with [tex]m\angle A=30[/tex]°AB = 2MCM is the mid-point of AB.Let AB = [tex]2x[/tex]Therefore, AM = MB = [tex]\frac{AB}{2}=x[/tex]Also, MC = [tex]\frac{AB}{2}=x[/tex]∴ AM = MB = MC = [tex]x[/tex]Now, consider triangle AMC,∵ AM = MC∴ [tex]m\angle MAC = \m\angle MCA = 30[/tex]°    ( [tex]m\angle A=m\angle MAC[/tex])Now, exterior angle BMC is given as the sum of opposite interior angles of triangle AMC.[tex]m\angle BMC=m\angle MAC+m\angle MCA\\m\angle BMC=30+30=60[/tex]Consider triangle BMC,∵ MB = MC∴ [tex]m\angle MBC = m\angle MCB = a(Let)[/tex]The sum of all interior angles is equal to 180°.[tex]m\angle BMC+m\angle MBC+m\angle MCB=180\\60+a+a=180\\2a=180-60\\2a=120\\a=\frac{120}{2}=60[/tex]Therefore, [tex]m\angle B =a = 60[/tex]°Also, [tex]m\angle C=m\angle MCA+m\angle MCB = 30+60=90[/tex]°Therefore, the triangle ABC is a special right angled triangle with measures 30° - 60° - 90°.For a special right angled triangle 30° - 60° - 90°, the hypotenuse is twice the base.Here, AB is the hypotenuse and BC is the base. So,[tex]AB=2BC\\AB=2\times 7=14\ ft[/tex]Therefore, AB = 14 ft.