We have three given points. We need to graph them, and then find the distances between them.
We need to remember that we can classify the triangles according to their sides:
1. A triangle with three congruent sides is an equilateral triangle.
2. A triangle with two congruent sides is an isosceles triangle.
3. A triangle with no congruent sides is a scalene triangle.
Additionally, we know that a segment is congruent to other when it has the same size as the other.
Then we can graph the three points as follows:
Now, we need to find the distances between the sides of the triangle using the distance formula as follows:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
This is the distance formula for points (x1, y1) and (x2, y2).
Finding the distance between points D and E
The coordinates for the two points are D(6, -6) and E(39,-12), and we can label them as follows:
• (x1, y1) = (6, -6) and (x2, y2) = (39, -12)
Then we have:
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ \\ d=\sqrt{(39-6)^2+(-12-(-6))^2} \\ \\ d=\sqrt{(33)^2+(-12+6)^2} \\ \\ d=\sqrt{33^2+(-6)^2}=\sqrt{1089+36}=\sqrt{1125} \\ \\ d_{DE}=\sqrt{1125}\approx33.5410196625 \end{gathered}[/tex]
Therefore, the distance between points D and E is √1125.
And we need to repeat the same steps to find the other distances.
Finding the distance between points E and F
We can proceed similarly as before:
The coordinates of the points are E(39, -12) and F(24, 18)
• (x1, y1) = (39, -12)
,
• (x2, y2) = (24, 18)
Then we have:
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ \\ d=\sqrt{(24-39)^2+(18-(-12))^2} \\ \\ d=\sqrt{(-15)^2+(18+12)^2}=\sqrt{(-15)^2+(30)^2}=\sqrt{225+900} \\ \\ d_{EF}=\sqrt{1125}\approx33.5410196625 \end{gathered}[/tex]
Then the distance between points E and F is √1125.
Finding the distance between F and D
The coordinates of the points are F(24, 18) and D(6, -6)
• (x1, y1) = (24, 18) and (x2, y2) = (6, -6)
Then we have:
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ \\ d=\sqrt{(6-24)^2+(-6-18)^2}=\sqrt{(-18)^2+(-24)^2}=\sqrt{324+576} \\ \\ d=\sqrt{900}=30 \\ \\ d_{FD}=30 \end{gathered}[/tex]
Now, we have the following measures for each of the sides of the triangle:
[tex]\begin{gathered} \begin{equation*} d_{DE}=\sqrt{1125}\approx33.5410196625 \end{equation*} \\ \\ d_{EF}=\sqrt{1125}\approx33.5410196625 \\ \\ d_{FD}=30 \end{gathered}[/tex]
Therefore, in summary, according to the results, we have two sides that are congruent (they have the same size). Therefore, the triangle DEF is an isosceles triangle.
