Coordinate Geometry 04 : geometry inside circles ⭕

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I am sorry for uploading Coord. 03 as well as Coord. 04 in this series in a single day. But the knowledge that I gave you {Coorg3 + Coorg4} would definitely come in handy during our further conquests.    👁️L👁️🕵️‍♂️

Circles

Circle is defined as a 2D shape in which every point on its edges or  circumference be located at the same equal distance from the center point, say O.

Circle and radial curves have always been loved by humanity : from islamic era forts to forts of Rajasthan, we find curves with different radius of curvature. We already told you in coorg 03 that by rounders we could draw many arcs of different radius. Due to different radius, the curvature of bending of arc changes.

Graphics of a circle

What interests me towards a circle

Radial curves are a reality in life. We never see any other geo-metrical shapes as much as we see circles. Circles are therefore a lot much of our interest. Moreover, we could place any triangle inside a circle : it called circumscribing the triangle. Let's see the fundamental axioms involved in circumscribing a triangle inside circle.

Axioms of circumscribing

4.1 : Equal chords AB and EV would cause a same angle on the center of circle

Also note that the perpendicular distance from O to EV (_say it be ON) and the perpendicular distance from O to AB is also equal (_say OM)

We conclude that in the same circle, if two chords are equal 
That is if(EV==AB) then ON=OM. ..(1)
   And  Angle(EOV) at O = Angle(AOB) at O ... (2)

Refer an example question below 👇 

Question says that consider “the length of chords AB and CD are equal then find the length { OM + ON } ”.

Solution to the question is 

Since AB and CD are chords of equal length of a same circle (important thing in a same circle is not the circle itself, even chords in two different circles with same radius have the valid application of our properties : we want same radius or radius of curvature)

Since AB=CD
Hence, ON=OM = say x
Now, ON+OM= x + x = 2x

4.2 : the line starting from center of circle to a chord acts as a perpendicular as well as bi-sector (2 equal divisions) to the chord

4.3 : a triangle inside a semicircle with its longest side as the hypotenuse is always a right angled triangle

triangl(ERV) is right angled at R

4.4 : angle subtended by a chord on the same side of circle, than angle at center will be twice to angle at circumference of same side of circle.

Ang(POQ) = Twice of Ang(PAQ)

Now, let's go for example questions

Question says points P (3,1), Q (8,3) and R (x, y) are three points such that triangle(PRQ) is a right angled at R and the area of triangle(PRQ) is 7 sw units always. Find the number of possible such points R.

Solution is a homework... We will discuss after next post is published...