In a right-angled triangle ABC, angle C = 90 degrees M midpoint AC N midpoint CB MN 6 cm angle

In a right-angled triangle ABC, angle C = 90 degrees M midpoint AC N midpoint CB MN 6 cm angle MNC = 60 degrees. Find the sides of triangle ABC. cos B. Area of triangle MNC.

Consider a right-angled triangle ABC with right angle C.

Points M and N are midpoints of legs AC and CB and MN = 6, angle MNC = 60 °.

MN is the midline of triangle ABC. Consequently,

MN = 1/2 * AB and MN is parallel to AB. From this we get

AB = 2 * MN = 2 * 6 = 12 and ABC angle = MNC angle = 60 °.

Then

AC = AB * sin (ABC) = 12 * sin (60 °) = 12 * √3 / 2 = 6 * √3.

BC = AB * cos (ABC) = 12 * cos (60 °) = 12 * 1/2 = 6.

Area S of triangle MNC:

S = 1/2 * CM * CN = 1/2 * AC / 2 * CB / 2 = AC * CB / 8 =

= 6 * 6 * √3 / 8 = 9 * √3 / 2.