Hello math bugs(🐞) & hivers(🐝)
I hope you are strong and stout and doing well in life.
Today I have come up with a circle and it's chords related problem.Many of us just cram the formulla and get the solution. You must have seen the problem in the cover photo. The problem is when two chords of a circle meet out side the circle and thus the chord divide d into four parts. You can see the four part are x,8,12 & 3 unit.We need to find the value of x as shown in the figure below:
The equation of the problem can be given by x(x+8)=12(3+12).Now we can find value of x easily but before finding the value, I want to show how to prove it.To prove it we have draw a little bit of construction.I know many of you got it how does the formulla come:
So we have to connect BD and CF as you can see in the following figure.This we find two similiar triangle namey ∆ABD and ∆ACE.
Let's see how both of the triangle become similiar.From ∆ABD and ACE , we find angle ABD and angle ACE equal as the are angle made by same arc on the circumference. Besides, angle BAC is the common angle for both of the triangle.Thus by A-A postulate of a triangle we the triangle similiar and so the third angle of the triangles angle AEC and angle ADB are equal.
So according to the similiar triangle property we find the following equation.If you have problem to find similiar angles or sides, you can check my previous article about similarity. I explained how to work on similarity in a convenient way.By the way I don't use similarity as similarity I think of trigonometry while finding solution related to it.Find similarity here
So you have seen how we get the formulla.Yeah similarity is very cool.Now it's time for finding the solution.Check the solution below.If you do not get it why and how the factor (x+18) can not be zero. It's because in reality a distance can be equal to zero. Hence the zero factor must be (x-10) and so x=10 is the solution. Check details below.
Some of you ask for me how I make all the figures. Here is the descripton. I use two android apps to prepare my images.
I hope you have found my article helpful.
Thanks you so much for comming so far.
Have a good day.
All is well.
