If the area of `triangle ABC` is 21, and the length of the height minus the length of the base equals 1, which of the following is equal to the base of the triangle ?

Options

A. `2`
B. `4`
C. `6`
D. `7`

Answer

To solve the problem step by step, we will use the information provided in the question regarding the area of triangle ABC and the relationship between the height and the base.

### Step 1: Define Variables
Let the base of the triangle be denoted as \( B \) and the height as \( H \).

### Step 2: Use the Area Formula
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the area is 21, we can write:
\[ \frac{1}{2} \times B \times H = 21 \]
Multiplying both sides by 2 gives:
\[ B \times H = 42 \quad \text{(Equation 1)} \]

### Step 3: Use the Relationship Between Height and Base
We are also given that the height minus the base equals 1:
\[ H - B = 1 \]
From this, we can express \( H \) in terms of \( B \):
\[ H = B + 1 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Now, substitute \( H \) from Equation 2 into Equation 1:
\[ B \times (B + 1) = 42 \]
Expanding this gives:
\[ B^2 + B = 42 \]

### Step 5: Rearrange the Equation
Rearranging the equation leads to:
\[ B^2 + B - 42 = 0 \]

### Step 6: Factor the Quadratic Equation
Next, we will factor the quadratic equation:
\[ B^2 + 7B - 6B - 42 = 0 \]
Grouping the terms gives:
\[ (B^2 + 7B) + (-6B - 42) = 0 \]
Factoring by grouping:
\[ B(B + 7) - 6(B + 7) = 0 \]
Factoring out \( (B + 7) \):
\[ (B + 7)(B - 6) = 0 \]

### Step 7: Solve for \( B \)
Setting each factor to zero gives:
\[ B + 7 = 0 \quad \text{or} \quad B - 6 = 0 \]
Thus, we find:
\[ B = -7 \quad \text{or} \quad B = 6 \]

### Step 8: Determine the Valid Base Length
Since the length of the base cannot be negative, we discard \( B = -7 \) and accept:
\[ B = 6 \]

### Conclusion
The base of the triangle is \( 6 \).