Find the perimeter of a triangle whose vertices pass through ( 3, 2), (4, 5) and (6, 2) in surd form

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 + \(\sqrt{10} + \sqrt{13}\).