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DISTANCIA ENTRE DOS PUNTOS EN EL PLANO. Para los ejercicios del 26 al 29. Un terreno rectangular tiene un perímetro de 360 m.

Let's solve the given exercises for the triangle \( ABC \) formed by the points \( A(-4, 1) \), \( B(1, 6) \), and \( C(3, -6) \).

### Problem 26: Calculate the lengths of the sides of triangle \( ABC \) and show that it is a right triangle.

We can find the lengths of the sides of the triangle using the distance formula:

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

For each side:

1. **Side \( AB \):**
   \[
   AB = \sqrt{(1 - (-4))^2 + (6 - 1)^2} = \sqrt{(1 + 4)^2 + (6 - 1)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
   \]

2. **Side \( BC \):**
   \[
   BC = \sqrt{(3 - 1)^2 + (-6 - 6)^2} = \sqrt{(3 - 1)^2 + (-12)^2} = \sqrt{2^2 + (-12)^2} = \sqrt{4 + 144} = \sqrt{148} = 2\sqrt{37}
   \]

3. **Side \( AC \):**
   \[
   AC = \sqrt{(3 - (-4))^2 + (-6 - 1)^2} = \sqrt{(3 + 4)^2 + (-6 - 1)^2} = \sqrt{7^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}
   \]

### Show that the triangle is right-angled:

To show that the triangle is a right triangle, we check if the Pythagorean theorem holds:
\[
AB^2 + AC^2 = BC^2
\]

1. \( AB^2 = (5\sqrt{2})^2 = 50 \)
2. \( AC^2 = (7\sqrt{2})^2 = 98 \)
3. \( BC^2 = (2\sqrt{37})^2 = 148 \)

Since:
\[
50 + 98 = 148
\]
The triangle satisfies the Pythagorean theorem, so it is a right triangle.

### Problem 27: Determine the perimeter of the triangle \( ABC \).

The perimeter is the sum of the lengths of the sides:
\[
P = AB + BC + AC = 5\sqrt{2} + 2\sqrt{37} + 7\sqrt{2}
\]
Combining the like terms:
\[
P = (5\sqrt{2} + 7\sqrt{2}) + 2\sqrt{37} = 12\sqrt{2} + 2\sqrt{37}
\]

Thus, the perimeter of the triangle is \( 12\sqrt{2} + 2\sqrt{37} \) units.

Would you like further details or clarification on any part of this? Here are some related questions:

1. How do we derive the distance formula between two points?
2. Can we use another method to check if the triangle is right-angled?
3. What is the significance of the Pythagorean theorem in determining right triangles?
4. How can we approximate the value of the perimeter?
5. How would the results change if the triangle were not right-angled?

**Tip:** For any triangle, you can confirm its type (right, acute, obtuse) by checking the relationship between the squares of the side lengths.

Calcule la longitud de los lados del triángulo ABC, y con estos resultados muestre que el triángulo ABC es rectángulo. Determine el perímetro del triángulo ABC.

Calculate the side lengths of triangle ABC with given points A(-4, 1), B(1, 6), and C(3, -6). Show that the triangle is right-angled using the Pythagorean theorem and determine the perimeter.

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