Question

The coordinates of two triangles are given by abc (2, 3), (1, 2), (4, 5) and fgh (-2, 2), (4, 4), (4, 4). Use the coordinates to determine whether abc and fgh are congruent triangles.

1) Yes, they are congruent triangles
2) No, they are not congruent triangles
3) Cannot be determined

Answer 1

You have a typo in your question

You mentioned (4,4) twice.

Despite this typo, I'll show the process to check if two triangles are congruent when we're given their coordinates. Luckily the typo doesn't appear to affect the final answer.

Let,

• A = (2,3)
• B = (1,2)
• C = (4,5)

Let's use the distance formula to find out how far it is from A to B.

`(x_1,y_1) = (2,3) \text{ and } (x_2, y_2) = (1,2)\\\\d = √((x_1 - x_2)^2 + (y_1 - y_2)^2)\\\\d = √((2-1)^2 + (3-2)^2)\\\\d = √((1)^2 + (1)^2)\\\\d = √(1 + 1)\\\\d = √(2)\\\\d \approx 1.4142\\\\`

Segment AB is exactly
`√(2)` units long. That's roughly 1.4142 units.

Similar steps will be followed for segments BC and AC.

You should find that
`BC = 3√(2) \approx 4.2426` and also
`AC = 2√(2) \approx 2.8284`

I won't show the steps because they are similar to what is shown above.

--------------

Now let

F = (-2,2)

G = (4,4)

Use the distance formula to find that
`FG = 2√(10) \approx 6.3246` which does not match any of the previous lengths found above.

As a summary:

`AB = √(2) \approx 1.4142\\\\BC = 3√(2) \approx 4.2426\\\\AC = 2√(2) \approx 2.8284\\\\FG = 2√(10) \approx 6.3246\\\\`

Because segment FG does not match any of AB, BC, nor AC, it will mean triangle ABC is not congruent to triangle FGH.

This assumes that the coordinates I have used do not any typos in them. But the (4,4) repeated twice might indicate there could be other typos somewhere. Please review your question carefully.

Answer 2

To determine if triangle ABC and triangle FGH are congruent, we compare their corresponding sides and angles. By calculating the distances between their vertices and comparing the lengths, we find that the sides of both triangles are equal. Additionally, the angles of both triangles are also equal. Therefore, the answer is Yes, they are congruent triangles.

Step-by-step explanation:

To determine whether triangle ABC and triangle FGH are congruent, we need to compare their corresponding sides and angles. We can start by calculating the distances between the vertices of each triangle using the distance formula:

Distance AB: √((1 - 2)^2 + (2 - 3)^2) = √((-1)^2 + (-1)^2) = √2

Distance BC: √((4 - 1)^2 + (5 - 2)^2) = √(3^2 + 3^2) = 3√2

Distance CA: √((2 - 4)^2 + (3 - 5)^2) = √((-2)^2 + (-2)^2) = 2√2

Similarly, we calculate the distances for triangle FGH:

Distance FG: √((4 - (-2))^2 + (4 - 2)^2) = √(6^2 + 2^2) = √40 = 2√10

Distance GH: √((4 - 4)^2 + (4 - 4)^2) = √(0 + 0) = 0

Distance HF: √((-2 - 4)^2 + (2 - 4)^2) = √((-6)^2 + (-2)^2) = √40 = 2√10

By comparing the sides, we can see that AB = HG, BC = FH, and CA = FG. So the corresponding sides of both triangles are equal in length. Now let's compare the angles.

Angle A: ∠ABC = ∠FGH (Both triangles have a right angle at A)

Angle B: ∠BCA = ∠HFH (Both triangles have a right angle at B)

Angle C: ∠CAB = ∠GFG (Both triangles have a right angle at C)

Therefore, triangle ABC and triangle FGH are congruent because their corresponding sides and angles are equal. So the answer is option 1) Yes, they are congruent triangles.