Check whether (5, −2), (6, 4) and (7, −2) are the vertices of an isosceles triangle.
Yes
To check if the given points are vertices of an isosceles triangle, we find the lengths of all three sides and see if any two are equal.
Let the points be \(A(5, -2)\), \(B(6, 4)\), and \(C(7, -2)\).
First, find the length of side \(AB\) using the distance formula: \( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Here, from \(A(5, -2)\) to \(B(6, 4)\): \( AB = \sqrt{(6 - 5)^2 + (4 - (-2))^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37} \).
Next, find the length of side \(BC\) from \(B(6, 4)\) to \(C(7, -2)\): \( BC = \sqrt{(7 - 6)^2 + (-2 - 4)^2} = \sqrt{1^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37} \).
Now, find the length of side \(AC\) from \(A(5, -2)\) to \(C(7, -2)\): \( AC = \sqrt{(7 - 5)^2 + (-2 - (-2))^2} = \sqrt{2^2 + 0^2} = \sqrt{4} = 2 \).
We observe that \( AB = \sqrt{37} \) and \( BC = \sqrt{37} \), so two sides are equal.
Since a triangle with two equal sides is an isosceles triangle, the points \((5, -2), (6, 4), (7, -2)\) do form an isosceles triangle.