The document describes dividing an equilateral triangle with sides of 1 meter into two triangles by randomly selecting a point on the base and connecting it to the opposite vertex. It then calculates the expected area of the larger of the two resulting triangles. The larger triangle's expected area is 0.389 square meters when the random point on the base is 0.9 meters from one end of the base.

1. Consider an equilateral triangle where the length of each side equals 1 meter. A point on the
base of the triangle is chosen at random. The triangle is divided into two triangles by joining the
said point to the vertex opposite to the base by a straight line. Find the expected area of the larger
of the two triangles.
Solution
let the triangle ABC has a base BC
take a point D on BC at a distance of x mts from B
BD=x
CD=1-x
now let us find area ABD
i.e 1/2*AB*BD*sinB
1/2*1*1-x*sin 60
(1-x)/2*3/2
equilalateral triangle area 3/4 a2
it is of 3/4 sq mt
area of ABD will be of (1-x)*3/4
area of ACD will be of 3/4-(1-x)*3/4
it is of 3/4*x
if x is o.9 then the are of larger triangle will be of 0.389 sq mts