A CUTE PROBLEM ON CIRCLE AND RECTANGLE

Hello math bugs(🐞) and hivers(🐝)
I hope you are strong and stout and doing well in life.

Well come to another episode of mathematics problem and solution. Today the problem is a combination of a semi-circle and a rectangle as you can see in the following figure.

Before solving I wanna mention the all the concepts that we gonna use to solve it. But before that you may give it a try if you like.

Required concepts:

✅1)A postulate: Radius drawn from the centre of a circle to the point of contact of a tangent of the same circle is always perpendicular to it.Check it below:

✅2)Similarity: Here we need to apply on right angle triangles created by the perpendicular drawn from the vertex of an another right angle triangle to its hypotenuse. Check the figure given below:

Check details of similarity HERE

✅3)Area of a rectangle: Of course you need to know how to find area of a rectangle ..lol. check it in the figure below:

Here most important concept is similarity. I will suggest you to check the link to know how easily similarity can be used. You neither need to rotate the triangles nor to memorise it.Once you done with it, the problem is almost solved.

SOLUTION:

So we have a relation between sides EC, FC and BC derived from similarity. Now need to twist the relation a little bit. Check FC is also written as diameter of the O centric semi-circle given in problem.So, what we have now is 2R×BC=EC². Here R equals to radius of the semi-circle and BC can be said the side of the rectangle ABCD. So , BC is length of the rectangle. So,The required length(BC) =EC²/2R and you may have noticed that Here Radius can be termed as Breadth. Why? It will be parallel to breadth of the rectangle. As you know two sides of a rectangle are perpendicular to each other and I already mentioned the postulate in the first concept(OP is perpendicular to BC or FC) discussed above.So, it's almost done.

   AREA OF THE RECTANGLE

    =(Lenght × Breadth) unit²

    =(EC²/2R × R) unit²

    = EC²/2 unit²

    =7²/2 cm² [ As EC = 7 cm given]

    = 49/2 cm²

    = 24.5 cm²

Find Radius, Length and Breadth in the figure below:

I hope now you have got this. Visit SIMILARITY to know how and from where the relation FC × BC = EC² is derived.

Lol I made a typing mistake in a frame😂. There may be other mistakes ;please ignore it if you can.

Thank you so much for your visit and support.

Have a good day

All is well

Regards: @meta007