Circle O has a radius of 5 cm. The length of a chord AB is 8 cm. Circle Q is tangent to a chord and to the arc AB as well. (See diagram below) If line segment AM is 2 cm, what's the area of a circle Q?

Dragan Dec 26, 2020

#1**+2 **

I haven't worked out how to do it with theoretical geometry and Algebra.

But here it is drawn to scale.

The radius is approx (maybe exact) 0.75

so the area is approx 0.5625pi cm^2

Melody Dec 27, 2020

#2**+3 **

I'll use your graph, if I may?!

Note that the **y-axis **is parallel to a chord **AB**, and the distance between them is **3 cm**.

The coordinates of the points P and O are: **P (0, 2)** **O (0, 0) **MD = ND ==> **r**

**PO = 2 PM = 3 ON = 5**

**PO ^{2} + (PM + r)^{2} = (ON - r)^{2}**

**2 ^{2} + (3 + r)^{2} = (5 - r)^{2} ==> r = 0.75**

**A = (3/4) ^{2}pi cm^{2}**

jugoslav
Dec 27, 2020

#3**+1 **

**That is a neat answer Jugoslav.**

Thanks, I was having a mental block.

The only thing you forgot is that it has to be shown that AB has a perpendicular distance to (0,0) of 3 units.

This is not hard to do, you can just use the Pythagorean triad 3-4-5 but it does have to be shown before your part of the calculations.

Melody
Dec 27, 2020