The lengths of the legs of a right-angled triangle ABC are 6 and 4 cm. The large leg AC is divided by points M and N

The lengths of the legs of a right-angled triangle ABC are 6 and 4 cm. The large leg AC is divided by points M and N into 3 equal parts. Points M and N are connected to vertex B. Find the area of triangle BMN.

Since, according to the condition, points M and H divide the AC leg into three equal segments, then

AM = MH = CH = AC / 3 = 2 cm.

Then AH = AM + MH = 2 + 2 = 4 cm.

The AНВ triangle is rectangular, since the angle is A = 90, then its area will be equal to:

Sanv = AН * AB / 2 = 4 * 4/2 = 8 cm2.

The AMB triangle is also rectangular, then Samv = AM * AB / 2 = 2 * 4/2 = 4 cm.

Svmn = Sanv – Samv = 8 – 4 = 4 cm2.

Answer: The area of the ВMН triangle is 4 cm2.