To find the perimeter of a triangle with vertices at given points, we need to calculate the lengths of its sides using the distance formula, which is given by:
Distance Formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
The vertices of the triangle are (3, 2), (4, 5), and (6, 2). Let's label these points as A(3, 2), B(4, 5), and C(6, 2).
Step 1: Finding AB
Using the distance formula for points A(3, 2) and B(4, 5):
\( AB = \sqrt{(4 - 3)^2 + (5 - 2)^2} \)
\( AB = \sqrt{1^2 + 3^2} \)
\( AB = \sqrt{1 + 9} \)
\( AB = \sqrt{10} \)
Step 2: Finding BC
Using the distance formula for points B(4, 5) and C(6, 2):
\( BC = \sqrt{(6 - 4)^2 + (2 - 5)^2} \)
\( BC = \sqrt{2^2 + (-3)^2} \)
\( BC = \sqrt{4 + 9} \)
\( BC = \sqrt{13} \)
Step 3: Finding CA
Using the distance formula for points C(6, 2) and A(3, 2):
\( CA = \sqrt{(3 - 6)^2 + (2 - 2)^2} \)
\( CA = \sqrt{(-3)^2 + 0^2} \)
\( CA = \sqrt{9} \)
\( CA = 3 \)
Step 4: Calculating the Perimeter
The perimeter of the triangle is the sum of the lengths of its sides:
\( \text{Perimeter} = AB + BC + CA \)
\( \text{Perimeter} = \sqrt{10} + \sqrt{13} + 3 \)
Thus, the perimeter of the triangle in surd form is 3 + \(\sqrt{10} + \sqrt{13}\).