Question

classify \\( \triangle a b c \\) by its sides. then determine whether it is a right triangle.
\\( a ( 2,3 ), b ( 6,3 ), c ( 2,7 ) \\)
the triangle can be classified by its sides as
this triangle a right triangle.

Explanation:

Step1: Calculate the lengths of the sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(AB\): \(x_1 = 2,y_1 = 3,x_2 = 6,y_2 = 3\)
\(AB=\sqrt{(6 - 2)^2+(3 - 3)^2}=\sqrt{4^2+0^2}=4\)
For \(AC\): \(x_1 = 2,y_1 = 3,x_2 = 2,y_2 = 7\)
\(AC=\sqrt{(2 - 2)^2+(7 - 3)^2}=\sqrt{0^2+4^2}=4\)
For \(BC\): \(x_1 = 6,y_1 = 3,x_2 = 2,y_2 = 7\)
\(BC=\sqrt{(2 - 6)^2+(7 - 3)^2}=\sqrt{(-4)^2+4^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\)

Step2: Classify the triangle by its sides

Since \(AB = AC=4\), the triangle is isosceles.

Step3: Check if it is a right - triangle using the Pythagorean theorem \(a^2 + b^2=c^2\) (where \(c\) is the longest side)

Let \(a = AB = 4,b = AC = 4,c = BC=4\sqrt{2}\)
\(a^2 + b^2=4^2+4^2=16 + 16=32\)
\(c^2=(4\sqrt{2})^2=32\)
Since \(a^2 + b^2=c^2\), it is a right - triangle.

Answer:

The triangle can be classified by its sides as isosceles. This triangle is a right triangle.