Certainly! Let's determine if the points [tex]\( D(-4,0) \)[/tex], [tex]\( E(-2,3) \)[/tex], and [tex]\( F(1,0) \)[/tex] form a right triangle by calculating the distances between the points and using the Pythagorean theorem.
### Step 1: Calculate the distances between each pair of points
#### Distance between [tex]\( D \)[/tex] and [tex]\( E \)[/tex]
The distance formula between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
For points [tex]\( D(-4,0) \)[/tex] and [tex]\( E(-2,3) \)[/tex]:
[tex]\[
\text{DE} = \sqrt{((-2) - (-4))^2 + (3 - 0)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606
\][/tex]
#### Distance between [tex]\( E \)[/tex] and [tex]\( F \)[/tex]
For points [tex]\( E(-2,3) \)[/tex] and [tex]\( F(1,0) \)[/tex]:
[tex]\[
\text{EF} = \sqrt{(1 - (-2))^2 + (0 - 3)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} \approx 4.243
\][/tex]
#### Distance between [tex]\( D \)[/tex] and [tex]\( F \)[/tex]
For points [tex]\( D(-4,0) \)[/tex] and [tex]\( F(1,0) \)[/tex]:
[tex]\[
\text{DF} = \sqrt{(1 - (-4))^2 + (0 - 0)^2} = \sqrt{(5)^2 + 0} = \sqrt{25} = 5.0
\][/tex]
### Step 2: Calculate the squares of these distances
[tex]\[
\text{DE}^2 \approx 3.606^2 = 13.0
\][/tex]
[tex]\[
\text{EF}^2 \approx 4.243^2 = 18.0
\][/tex]
[tex]\[
\text{DF}^2 = 5.0^2 = 25.0
\][/tex]
### Step 3: Check for the Pythagorean theorem
A set of points form a right triangle if the sum of the squares of two sides is equal to the square of the third side:
[tex]\[
\text{DE}^2 + \text{EF}^2 = \text{DF}^2 \quad \text{or} \quad \text{DE}^2 + \text{DF}^2 = \text{EF}^2 \quad \text{or} \quad \text{DF}^2 + \text{EF}^2 = \text{DE}^2
\][/tex]
Let's test these conditions:
[tex]\[
13.0 + 18.0 \neq 25.0
\][/tex]
[tex]\[
13.0 + 25.0 \neq 18.0
\][/tex]
[tex]\[
25.0 + 18.0 \neq 13.0
\][/tex]
### Conclusion
Since none of these conditions hold true, the points [tex]\( D(-4,0) \)[/tex], [tex]\( E(-2,3) \)[/tex], and [tex]\( F(1,0) \)[/tex] do not form a right triangle.
Therefore, the answer is No.
