What type of triangle is formed by joining the points D(7, 3), E(8, 1), and F(4, -1)?

Solution:

Given three points are:

D(7, 3), E(8, 1), and F(4, -1).

Using distance formula, d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Length DE = √[(8 - 7)² + (1 - 3)²]

= √(1 + 4)

= √5

Length EF = √[(4 - 8)² + (-1 - 1)²]

= √(16 + 4)

= √20

= √(4 × 5)

= 2√5

Length FD = √[(7 - 4)² + (3 - (-1))²

= √(9 + 16)

= √25

= 5

Using Pythagoras theorem to check if the triangle is right angled, which states that “the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse”.

DE² + EF² = (√5)² + (2√5)²

= 5 + 20

= 25

= FD²

Therefore, given vertices(points) form right angled triangle with right angle at vertex E.

Aliter

The slope of the line joining two points (x₁, y₁) and (x₂, y₂) is (y₂ - y₁)/(x₂ - x₁)

Slope of DE = (1 - 3)/ (8 - 7) = -2

Slope of EF = (-1 -1)/ (4 - 8) = 1/2

We find that,

(Slope of DE) × (Slope of EF) = -2 × 1/2 = -1

Therefore, DE ⟂ EF

Hence, DEF forms a right angled triangle.

---

What type of triangle is formed by joining the points D(7, 3), E(8, 1), and F(4, -1)?

Summary:

The triangle is formed by joining the points D(7, 3), E(8, 1), and F(4, -1) is a right angled triangle.