In a `angleABC`, `angleA` : `angleB` : `angleC` = 2 : 4 : 3. The shortest side and the longest side of the triangle are respectively

Options

A. AC and AB
B. AC and BC
C. BC and AC
D. AB and AC

Answer

To solve the problem, we need to find the shortest and longest sides of triangle ABC based on the given angles. Here’s a step-by-step solution:

### Step 1: Understand the Ratio of Angles
The angles of triangle ABC are given in the ratio:
- Angle A : Angle B : Angle C = 2 : 4 : 3

### Step 2: Assign Variables to Angles
Let’s assign a variable \( k \) to represent the common factor:
- Angle A = \( 2k \)
- Angle B = \( 4k \)
- Angle C = \( 3k \)

### Step 3: Use the Triangle Angle Sum Property
The sum of the angles in any triangle is always 180 degrees. Therefore, we can write the equation:
\[ 2k + 4k + 3k = 180 \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 9k = 180 \]

### Step 5: Solve for \( k \)
To find \( k \), divide both sides by 9:
\[ k = \frac{180}{9} = 20 \]

### Step 6: Calculate Each Angle
Now substitute \( k \) back into the expressions for the angles:
- Angle A = \( 2k = 2 \times 20 = 40^\circ \)
- Angle B = \( 4k = 4 \times 20 = 80^\circ \)
- Angle C = \( 3k = 3 \times 20 = 60^\circ \)

### Step 7: Identify the Shortest and Longest Angles
Now we have:
- Angle A = 40 degrees
- Angle B = 80 degrees
- Angle C = 60 degrees

The smallest angle is Angle A (40 degrees) and the largest angle is Angle B (80 degrees).

### Step 8: Determine the Corresponding Sides
According to the properties of triangles:
- The shortest side is opposite the smallest angle.
- The longest side is opposite the largest angle.

Thus:
- The shortest side (opposite Angle A) is side BC.
- The longest side (opposite Angle B) is side AC.

### Conclusion
The shortest side is BC and the longest side is AC.