Question

Are Triangle ABC congruent to triangle GHE where A (1,-3); B (1,-1); C (4, -3); E ( -2,5); H( -4,2); G(-2, 2)

Answer 1

To determine if triangle ABC is congruent to triangle GHE, we need to compare the lengths of their sides. We will first calculate the lengths of the sides of triangle ABC and triangle GHE using the distance formula.

The distance formula between points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate the lengths of sides in triangle ABC

**Length AB:**
\[
A(1, -3), B(1, -1)
\]
\[
AB = \sqrt{(1 - 1)^2 + (-1 - (-3))^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2
\]

**Length BC:**
\[
B(1, -1), C(4, -3)
\]
\[
BC = \sqrt{(4 - 1)^2 + (-3 - (-1))^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

**Length CA:**
\[
C(4, -3), A(1, -3)
\]
\[
CA = \sqrt{(1 - 4)^2 + (-3 - (-3))^2} = \sqrt{(-3)^2 + 0} = \sqrt{9} = 3
\]

### Step 2: Calculate the lengths of sides in triangle GHE

**Length GH:**
\[
G(-2, 2), H(-4, 2)
\]
\[
GH = \sqrt{(-4 - (-2))^2 + (2 - 2)^2} = \sqrt{(-2)^2 + 0} = \sqrt{4} = 2
\]

**Length HE:**
\[
H(-4, 2), E(-2, 5)
\]
\[
HE = \sqrt{(-2 - (-4))^2 + (5 - 2)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

**Length EG:**
\[
E(-2, 5), G(-2, 2)
\]
\[
EG = \sqrt{(-2 - (-2))^2 + (2 - 5)^2} = \sqrt{0 + (-3)^2} = \sqrt{9} = 3
\]

### Step 3: Compare the lengths of the sides

- Triangle ABC:
- \(AB = 2\)
- \(BC = \sqrt{13}\)
- \(CA = 3\)

- Triangle GHE:
- \(GH = 2\)
- \(HE = \sqrt{13}\)
- \(EG = 3\)

### Conclusion

Since the corresponding side lengths of triangle ABC and triangle GHE are equal:

- \(AB \equiv GH\)
- \(BC \equiv HE\)
- \(CA \equiv EG\)

We can conclude that triangle ABC is congruent to triangle GHE.

**Answer: Yes, triangle ABC is congruent to triangle GHE.**