Let `D` be the middle point of the side `B C` of a triangle `A B Cdot` If the triangle `A D C` is equilateral, then `a^2: b^2: c^2` is equal to

Options

A. `1:4:3`
B. `4:1:3`
C. `4:3:1`
D. `3:4:1`

Answer

To solve the problem step by step, we will analyze the triangle ABC with D as the midpoint of side BC and triangle ADC as an equilateral triangle.

### Step 1: Define the sides of triangle ABC
Let:
- AC = b = x (since we will denote the sides of triangle ABC as a, b, c)
- AD = c = x (since triangle ADC is equilateral)
- DC = BD = a/2 (since D is the midpoint of BC)

### Step 2: Understand the properties of triangle ADC
Since triangle ADC is equilateral, all sides are equal:
- AC = AD = DC = x

### Step 3: Identify the angles in triangle ADB
In triangle ADB:
- Angle ADB = 120° (since angle ADB is the exterior angle to angle ADC, which is 60°)

### Step 4: Apply the Cosine Rule in triangle ADB
Using the cosine rule:
\[ AB^2 = AD^2 + BD^2 - 2 \cdot AD \cdot BD \cdot \cos(120°) \]
Substituting the known values:
\[ c^2 = x^2 + \left(\frac{a}{2}\right)^2 - 2 \cdot x \cdot \frac{a}{2} \cdot \left(-\frac{1}{2}\right) \]
This simplifies to:
\[ c^2 = x^2 + \frac{a^2}{4} + \frac{ax}{2} \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ c^2 = x^2 + \frac{a^2}{4} + \frac{ax}{2} \]
We know that \(c = x\), so substituting \(c\):
\[ x^2 = x^2 + \frac{a^2}{4} + \frac{ax}{2} \]
This leads to:
\[ 0 = \frac{a^2}{4} + \frac{ax}{2} \]

### Step 6: Solve for a
Multiplying through by 4 to eliminate the fraction:
\[ 0 = a^2 + 2ax \]
Factoring gives:
\[ a(a + 2x) = 0 \]
Thus, \(a = 0\) or \(a = -2x\). Since side lengths cannot be negative, we discard \(a = 0\).

### Step 7: Find the ratio of the sides
Now we have:
- a = 2x (since D is the midpoint of BC)
- b = x
- c = x

Now we can find the ratios:
\[ a^2 : b^2 : c^2 = (2x)^2 : x^2 : x^2 = 4x^2 : x^2 : x^2 \]
This simplifies to:
\[ 4 : 1 : 1 \]

### Final Answer
Thus, the ratio \(a^2 : b^2 : c^2\) is equal to \(4 : 1 : 1\).