Question

Calculate the area of triangle CDE with altitude EF, given C (3, −2), D (−1, 2), E (2, 3), and F (0, 1).

4 square units
6.2 square units
8 square units
8.7 square units

Answer 1

To calculate the area of triangle CDE with altitude EF, we first need to find the base CD and the height EF.

1. **Calculate the length of base CD**:
The coordinates of points C and D are given as C(3, -2) and D(-1, 2).

The length of CD can be calculated using the distance formula:
\[
CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1) = C(3, -2)\) and \((x_2, y_2) = D(-1, 2)\).

\[
CD = \sqrt{((-1) - 3)^2 + (2 - (-2))^2}
\]
\[
= \sqrt{(-4)^2 + (4)^2}
\]
\[
= \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]

2. **Calculate the area of triangle CDE**:
The area \(A\) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here the base is CD and the height is the distance from point E to line segment CD.

3. **Find the equation of the line through points C and D**:
The slope \(m\) of line CD is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-2)}{-1 - 3} = \frac{4}{-4} = -1
\]
Using point-slope form to find the line equation through point C(3, -2):
\[
y + 2 = -1(x - 3)
\]
Rearranging, we get:
\[
y = -x + 1
\]

4. **Calculate the distance from point E(2, 3) to line CD**:
The distance \(d\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
From the line equation \(y = -x + 1\), we rewrite it in standard form:
\[
x + y - 1 = 0
\]
So, \(A = 1\), \(B = 1\), and \(C = -1\). Now substituting point E(2, 3):
\[
d = \frac{|1 \cdot 2 + 1 \cdot 3 - 1|}{\sqrt{1^2 + 1^2}} = \frac{|2 + 3 - 1|}{\sqrt{2}} = \frac{|4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}
\]

5. **Calculate the area of triangle CDE**:
Now substituting the base and height into the area formula:
\[
A = \frac{1}{2} \times (4\sqrt{2}) \times (2\sqrt{2})
\]
\[
= \frac{1}{2} \times 4 \times 2 \times 2 = \frac{16}{2} = 8 \text{ square units}
\]

Thus, the area of triangle CDE is \( \boxed{8} \) square units.