When they will be Equal

Geometry Level 4

Consider two straight lines $L_1 : y=x$ and $L_2 : x-y+3=0$ in X-Y plane.

Construct a triangle $T_1$ with the points $A=(0,0) ; B=(3,3);C=(2,5)$ .

Again construct another triangle $T_2$ with points $A,B$ and third vertex on the line $L_2$ .This third point is chosen randomly and independently on $L_2$ .

What is the probability that area of $T_1$ will be equal to area of $T_2$ ?

Submit your answer in three decimal places

The answer is 1.000.

1 solution

Kushal Bose
Jun 9, 2017

Note that slope of the two lines are equal.So, they are parallel lines.

It can be checked that points $A,B$ lies on $L_1$ and point $C$ lies on $L_2$ .So, triangle $T_1$ lies within the parallel lines.

For triangle $T_2$ it has common base with $T_1$ and third vertex lies also on the $L_2$ .

So, remembering a theorem that "If any two triangle lies with two parallel lines with a common base then choosing of any third vertex on the opposite line then these two triangles have both equal areas." .

Here, choosing of third vertex on the line $L_2$ will always make two areas equal.

The any point on $L_2$ will serve the matter.So, required probability is $1.000$

When they will be Equal

The answer is 1.000.

1 solution

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