N△abc points m and q are points on the sides ab and bc respectively, so that am=2mb and bq=qc. find area of quadrilateral mbqp if aq ∩ cm =p and aabc=30 cm2.

Denote the area of quadilateral MBQP as A, the area of the triangle MBP as and the area of the triangle BPQ as

Note that:

1.

2.

3.

4.

Now

You get the system of three equations:

Substitute the first equation into the last two:

From the first equation and then

Thus,

and

7 sq. cm

Math has the particularity that it is a logical construction.

This means that we can start with an expression X (where X is an equation, not a variable)

Now we can apply a lot of "math" to this equation in such a way that we can rewrite it, but the actual "meaning" of the equation will not change.

An example of this is factoring.

For example, we can write a quadratic equation as:

a*x^2 + b*x + c.

And we also can write this as:

n*(x - k)*(x - j)

where k and j are the solutions of the equation:

a*x^2 + b*x + c  = 0.

What is the advantage of writing the equation in each form?

Well, both expressions actually represent the same thing, but the explicit information in each expression is different, so depending on what we want to do, we will choose one option or the other.

And we have lot's of different ways to express something, where we can find some ones really complex and useful, like the series of Taylor, where we can write a function as a summation of infinite terms.