Finding Area of Triangle Inside a Triangle made by Two Cir-cum-raidus and a side.

Hello math bugs(🐞) & hivers(🐝)

Well come to another interesting geometric concept and Solution of the given problem with the help of that.

You have already seen the problem figure. It may seems hard but with proper understanding of a few concepts, it can be solved very easily. Before starting, let me elaborate first what the problem is asking.

There is Large triangle ∆ABC circumscribed by a circle with centre O and if AO and BO joined , we get another samller triangle called ∆ABO. And we need to find out the area of it.

Concepts/ postulates/ axioms what we need to solve it are given below👇

1) Heron's Formula:

It says if the three sides of a trinagle are a, b and c, then area can be given by

Ar. of ∆ = root over of [S(S-a)(S-b)(S-c)] unit²
Where S= semi- perimeter of ∆= (a+b+c)/2
and a, b & c are three side of the ∆

Here is how Heron's Formula can be obtained.

2) Formula of Cir-cum-raidus:

Cir-cum-raidus (R) = (abc)/(4∆) unit.

Where, R = Cir-cum-raidus.
a, b & c are three sides of a triangle.
∆ = area of triangle.

Here is how Cir-cum-raidus can be given by so.

3) The Pythagoras Formula:

Hypotensue² = Base² + perpendicular²

Know about Pythagorean Triplet.

4) Area of triangle:

Area of triangle(∆)= (1/2)× Base× height unit²
This is the basic formula, you already known to it.

I haven't shared it here on hive. If you want me to prove it, hit a comment.

The Solution:

We are going to use the above four formulas in the same order to draft the final answer.

Using the first formula we can have the area of ∆ABC.
So, area of ∆ABC = 84 cm². How! check it below: 👇

Now we can find value of Cir-cum-raidus(R) using second formula. So, R = 65/8 cm. How! Check it below: 👇

Then using the third formula, we can find height (H) of the ∆ABO. H = 39/8 cm. How check it below: 👇

As we have got the height of ∆ABO, now it is very easy to find area of it. So, area of ∆ABO = 507/16 cm². How! Check it below: 👇

I hope my today's work is likeable. Add me in your favourite author list to get notification or keep visiting once in a week for more intersting mathamatics and geometric facts & concepts.

📢📢All the figures / pictures are made by me using android application. There may be some silly mistakes, though I tried my best to check them out. If still there is any, excuse me for that and please try to ignore it. And figures may not be accurate in measurement, consider the given data only.

Thank you so much for stopping by.

Have a nice day.

All is well.

Regards: @meta007