To find the area of the triangle inscribed in the parabola \( y^2 = 4x \) with vertices having ordinates 1, 2, and 4, we can follow these steps:
### Step 1: Identify the coordinates of the vertices
Given the ordinates (y-coordinates) of the vertices are \( y_1 = 1 \), \( y_2 = 2 \), and \( y_3 = 4 \). We need to find the corresponding x-coordinates using the parabola equation \( y^2 = 4x \).
### Step 2: Calculate the x-coordinates
Using the equation \( y^2 = 4x \):
1. For \( y_1 = 1 \):
\[
1^2 = 4x_1 \implies 1 = 4x_1 \implies x_1 = \frac{1}{4}
\]
So, the first vertex \( A \) is \( \left( \frac{1}{4}, 1 \right) \).
2. For \( y_2 = 2 \):
\[
2^2 = 4x_2 \implies 4 = 4x_2 \implies x_2 = 1
\]
So, the second vertex \( B \) is \( (1, 2) \).
3. For \( y_3 = 4 \):
\[
4^2 = 4x_3 \implies 16 = 4x_3 \implies x_3 = 4
\]
So, the third vertex \( C \) is \( (4, 4) \).
### Step 3: Write down the coordinates of the triangle vertices
The vertices of the triangle are:
- \( A \left( \frac{1}{4}, 1 \right) \)
- \( B (1, 2) \)
- \( C (4, 4) \)
### Step 4: Use the formula for the area of a triangle
The area \( A \) of a triangle given vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[
A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
### Step 5: Substitute the coordinates into the area formula
Substituting the coordinates:
\[
A = \frac{1}{2} \left| \frac{1}{4}(2 - 4) + 1(4 - 1) + 4(1 - 2) \right|
\]
Calculating each term:
1. \( \frac{1}{4}(2 - 4) = \frac{1}{4}(-2) = -\frac{1}{2} \)
2. \( 1(4 - 1) = 1 \cdot 3 = 3 \)
3. \( 4(1 - 2) = 4(-1) = -4 \)
Now substituting back:
\[
A = \frac{1}{2} \left| -\frac{1}{2} + 3 - 4 \right| = \frac{1}{2} \left| -\frac{1}{2} - 1 \right| = \frac{1}{2} \left| -\frac{3}{2} \right| = \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}
\]
### Final Answer
The area of the triangle inscribed in the parabola is \( \frac{3}{4} \) square units.
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