Triangle ABC has coordinates A(-4,-2), B(0,-2), and C(-4,1).

Triangle DEF has coordinates D(3,6), E(3,2), and F(0,2).

Use coordinate geometry to determine if these two triangles are congruent. If they are not congruent, state the reason.
A: The triangles are congruent.

B: The triangles are not congruent because corresponding angles are not congruent.

C: The triangles are similar but they are not congruent.

D: The triangles are not congruent because corresponding sides are not congruent.

Can someone please teach me how to do this

You could go through the math of finding the side lengths of all six sides, and comparing them, but these are easy to figure out if you just plot the points.

Then its easy to see that ABC is a 3-4-5 right triangle.

Now take a look at DEF.

Ccw

Sure! In order to determine if two triangles are congruent, we need to compare their corresponding sides and angles.

For Triangle ABC, let's calculate the lengths of its sides:

Side AB:
Distance formula: sqrt((0-(-4))^2 + (-2-(-2))^2) = sqrt(16+0) = sqrt(16) = 4

Side BC:
Distance formula: sqrt((-4-0)^2 + (1-(-2))^2) = sqrt((-4)^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5

Side AC:
Distance formula: sqrt((-4-(-4))^2 + (1-(-2))^2) = sqrt(0^2 + 3^2) = sqrt(9) = 3

Now, let's calculate the lengths of Triangle DEF.

Side DE:
Distance formula: sqrt((3-3)^2 + (6-2)^2) = sqrt(0 + 16) = sqrt(16) = 4

Side EF:
Distance formula: sqrt((0-3)^2 + (2-2)^2) = sqrt((-3)^2 + 0) = sqrt(9) = 3

Side FD:
Distance formula: sqrt((0-3)^2 + (2-6)^2) = sqrt((-3)^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5

By comparing the lengths of the sides, we can see that Triangle ABC has sides of lengths 4, 5, and 3, while Triangle DEF has sides of lengths 4, 3, and 5. Since the corresponding sides of the two triangles are congruent, we can conclude that the triangles are congruent.

So, the correct answer is A: The triangles are congruent.

To determine if two triangles are congruent using coordinate geometry, we need to compare their corresponding sides and corresponding angles.

First, we can calculate the lengths of the sides of triangle ABC:
- The length of AB can be found using the distance formula: sqrt((0-(-4))^2 + (-2-(-2))^2) = 4
- The length of BC can also be found using the distance formula: sqrt((-4-0)^2 + (1-(-2))^2) = 5
- The length of AC can be found using the distance formula: sqrt((-4-(-4))^2 + (1-(-2))^2) = 3

Next, we can calculate the lengths of the sides of triangle DEF:
- The length of DE can be found using the distance formula: sqrt((3-3)^2 + (6-2)^2) = 4
- The length of EF can also be found using the distance formula: sqrt((0-3)^2 + (2-2)^2) = 3
- The length of DF can be found using the distance formula: sqrt((0-3)^2 + (2-6)^2) = 5

Comparing the corresponding side lengths, we can see that AB is congruent to DE, BC is congruent to EF, and AC is congruent to DF. Therefore, the corresponding sides of the two triangles are congruent.

Now, let's compare the corresponding angles in the two triangles.
- Angle A in triangle ABC is formed by the points A(-4,-2), B(0,-2), and C(-4,1).
- Angle D in triangle DEF is formed by the points D(3,6), E(3,2), and F(0,2).

To compare the angles, we can calculate the slopes of the two sides of the angles:
- The slope of side AB in triangle ABC is (change in y / change in x) = (-2-(-2)) / (-4-0) = 0/4 = 0.
- The slope of side DE in triangle DEF is (change in y / change in x) = (6-2) / (3-3) = 4/0, which is undefined.

Since the slopes of the sides are different, the corresponding angles are not congruent.

Therefore, the correct answer is D: The triangles are not congruent because corresponding sides are not congruent.

To determine if two triangles are congruent using coordinate geometry, we need to compare their corresponding sides and corresponding angles.

First, let's calculate the lengths of the sides of Triangle ABC:

Side AB:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - (-4))^2 + (-2 - (-2))^2]
= √[(4)^2 + (0)^2]
= √[16 + 0]
= √16
= 4

Side AC:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(-4 - (-4))^2 + (1 - (-2))^2]
= √[(0)^2 + (3)^2]
= √[0 + 9]
= √9
= 3

Side BC:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - (-4))^2 + (-2 - 1)^2]
= √[(4)^2 + (-3)^2]
= √[16 + 9]
= √25
= 5

Now, let's calculate the lengths of the sides of Triangle DEF:

Side DE:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(3 - 3)^2 + (2 - 6)^2]
= √[(0)^2 + (-4)^2]
= √[0 + 16]
= √16
= 4

Side DF:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - 3)^2 + (2 - 2)^2]
= √[(-3)^2 + (0)^2]
= √[9 + 0]
= √9
= 3

Side EF:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(0 - 3)^2 + (2 - 6)^2]
= √[(-3)^2 + (-4)^2]
= √[9 + 16]
= √25
= 5

Now, let's compare the corresponding sides:

Side AB (4) is not equal to Side DE (4).
Side AC (3) is equal to Side DF (3).
Side BC (5) is equal to Side EF (5).

Since at least two pairs of corresponding sides are equal, we can conclude that the two triangles are congruent.

Therefore, the correct answer is A: The triangles are congruent.