How can I find the surface area of the circle m only with the given information?

Question

I have this question in my textbook: AB is a chord in the circle M, MC is perpendicular to AB, D is the midpoint of MA, CD = 3 cm, then the surface area of the circle M = how many squared centimeters?
I tried to solve it by getting the radius of the circle, but I can't get it from the provided information. From the model answers, it seems that the right radius length should be 6, double the length of CD, but I still can't prove it should be so from the question. Could you please help me?

Answer 1

Theorem: The median of a triangle to a side is half of the side in length if and only if the triangle is a right-angled triangle and the side is the hypotenuse.
Proof: We will first prove that the median to the hypotenuse is half of it. Consider a right-angled $\triangle ABC$ where $AC$ is the hypotenuse and $BD$ is the median drawn on it. Construct a line passing from $D$ parallel to $BC$ which cuts $AB$ at $E$.

$$\angle AED = \angle ABC = 90^\circ = \angle BED \tag{Corresponding angles}\label{1}$$ $$AD = CD \tag{$BD$ is median}\label{2}$$ $$AE = BE \tag{Converse of Mid-Point Theorem}\label{3}$$ Consider $\triangle BED$ and $\triangle AED$: $$AE = BE \tag{shown above}\label{4}$$ $$\angle AED = \angle BED \tag{shown above}\label{5}$$ $$ED = ED \tag{Common}\label{6}$$ Thus, by $SAS$ rule of congruency, we get $\triangle BED \cong \triangle AED$ and hence $BD = AD$. But we also have $AD = CD$, hence, $$\boxed{BD = AD = CD = \frac 12 AC}$$

Now, we will show that if in a triangle, the median drawn to a side is equal to half of it in length, the triangle is a right-angled triangle, and the angle opposite to the longest side is $90^\circ$ (or the side is the hypotenuse). Consider $\triangle ABC$, where $BD$ is a median drawn to $AC$ which is half in length of $AC$.

$$AD = CD = \frac 12 AC\tag{$BD$ is median}\label{7}$$ $$\color{blue}{BD} = \frac 12 AC = \color{blue}{AD} = \color{blue}{CD} \tag{Given $BD = 0.5 AC$}\label{8}$$ So, we can draw a circle centred at $D$ passing through $A,B,C$.

Clearly, $AC$ is the diameter, and by Thales's Theorem$^{[*]}$, $$\boxed{\angle ABC = 90^\circ \iff \triangle ABC \text{ is right-angled, and hypotenuse is $AC$}}$$ QED.

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Using this theorem, you can easily show that the radius of the circle in question is $6$ cm.

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$^{[*]}$: https://en.wikipedia.org/wiki/Thales%27s_theorem

Answer 2

Draw a line from $D$ to some point $D'$ on $CA$ such that the line connecting $D$ to $D'$ is perpendicular to $CA$. Using the concept of Similar Triangles, you should be able to show that the triangles $DCD'$ and $DAD'$ are the "same" triangle (you should try and prove this for yourself, it'll be a good practice). This implies that $$\text{Length of }DA \;=\; \text{Length of }DC \;=\; 3.$$ You should be able to determine the radius of the circle with that.