Classifying a Triangle Using Vertices

To classify a triangle given its vertices, we need to determine the lengths of its sides. The side lengths allow us to check if the triangle is isosceles (two sides equal), equilateral (all sides equal), or scalene (all sides different). We can also check if it's a right-angled triangle using the Pythagorean theorem.

Let the vertices of the triangle be A(3,1), B(-1,0), and C(2,5).

Calculating Side Lengths with the Distance Formula

The distance between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ in a coordinate plane is given by the distance formula:

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

Length of Side AB

Using points A(3,1) and B(-1,0):

$$ AB = \sqrt{(-1 - 3)^2 + (0 - 1)^2} $$

$$ AB = \sqrt{(-4)^2 + (-1)^2} $$

$$ AB = \sqrt{16 + 1} $$

$$ AB = \sqrt{17} $$

Length of Side BC

Using points B(-1,0) and C(2,5):

$$ BC = \sqrt{(2 - (-1))^2 + (5 - 0)^2} $$

$$ BC = \sqrt{(2 + 1)^2 + 5^2} $$

$$ BC = \sqrt{3^2 + 5^2} $$

$$ BC = \sqrt{9 + 25} $$

$$ BC = \sqrt{34} $$

Length of Side AC

Using points A(3,1) and C(2,5):

$$ AC = \sqrt{(2 - 3)^2 + (5 - 1)^2} $$

$$ AC = \sqrt{(-1)^2 + 4^2} $$

$$ AC = \sqrt{1 + 16} $$

$$ AC = \sqrt{17} $$

Checking for Isosceles Triangle

A triangle is isosceles if at least two of its sides have equal length.

We found the lengths of the sides to be:

• AB = $ \sqrt{17} $
• BC = $ \sqrt{34} $
• AC = $ \sqrt{17} $

Since AB = AC ($ \sqrt{17} $), the triangle has two sides of equal length. Therefore, the triangle is an isosceles triangle.

Checking for Right-Angled Triangle using Pythagorean Theorem

A triangle is right-angled if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides ($a^2 + b^2 = c^2$). The longest side is opposite the right angle.

The side lengths are $ \sqrt{17} $, $ \sqrt{34} $, and $ \sqrt{17} $. The longest side is BC with length $ \sqrt{34} $. Let's square the lengths of all sides:

• $ AB^2 = (\sqrt{17})^2 = 17 $
• $ BC^2 = (\sqrt{34})^2 = 34 $
• $ AC^2 = (\sqrt{17})^2 = 17 $

Now, let's check if the sum of the squares of the two shorter sides equals the square of the longest side:

$$ AB^2 + AC^2 = 17 + 17 = 34 $$

We see that $ AB^2 + AC^2 = 34 $, which is equal to $ BC^2 $. This satisfies the Pythagorean theorem.

Therefore, the triangle is a right-angled triangle.

Conclusion: Classifying the Triangle

Based on our calculations, the triangle with vertices (3,1), (-1,0), and (2,5) is:

• Isosceles: because two sides (AB and AC) are equal in length ($ \sqrt{17} $).
• Right-angled: because the square of the longest side (BC) is equal to the sum of the squares of the other two sides ($ BC^2 = AB^2 + AC^2 $).

Thus, the triangle is both isosceles and right-angled.

Revision Table: Key Geometric Concepts

Additional Information: Types of Triangles

Triangles can be classified based on their side lengths and angles:

• By Side Lengths:
  • Scalene: All three sides have different lengths.
  • Isosceles: At least two sides have equal lengths.
  • Equilateral: All three sides have equal lengths.
• By Angles:
  • Acute: All three angles are less than 90 degrees.
  • Right-Angled: One angle is exactly 90 degrees.
  • Obtuse: One angle is greater than 90 degrees.

A triangle can have characteristics from both classifications, such as being isosceles and right-angled, as seen in this problem.