Question 4

Isosceles Triangle: A triangle is called isosceles if exactly two of its three sides are equal in length. To verify, we calculate the distance between each pair of points and check if at least two distances are equal.

Distance Formula: The distance between two points $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

We will use the distance formula to find the length of each side of the triangle.

Step 1 — Calculate the length of side AB

Let's label the given points as A($\mathbf{5, -2}$), B($\mathbf{6, 4}$), and C($\mathbf{7, -2}$). We use the distance formula: $\sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$.

$AB = \sqrt{((6 - 5)^2 + (4 - (-2))^2)}$

$= \sqrt{((1)^2 + (4 + 2)^2)}$

$= \sqrt{(1^2 + 6^2)}$

$= \sqrt{(1 + 36)}$

$\boxed{AB = \sqrt{37} \text{ units}}$

Step 2 — Calculate the length of side BC

Now, let's find the distance between points B($\mathbf{6, 4}$) and C($\mathbf{7, -2}$). We apply the distance formula again.

$BC = \sqrt{((7 - 6)^2 + (-2 - 4)^2)}$

$= \sqrt{((1)^2 + (-6)^2)}$

$= \sqrt{(1 + 36)}$

$\boxed{BC = \sqrt{37} \text{ units}}$

Step 3 — Calculate the length of side CA

Next, we find the distance between points C($\mathbf{7, -2}$) and A($\mathbf{5, -2}$). We use the distance formula one more time.

$CA = \sqrt{((5 - 7)^2 + (-2 - (-2))^2)}$

$= \sqrt{((-2)^2 + (-2 + 2)^2)}$

$= \sqrt{((-2)^2 + (0)^2)}$

$= \sqrt{(4 + 0)}$

$= \sqrt{4}$

$\boxed{CA = 2 \text{ units}}$

Step 4 — Check if it is an isosceles triangle

We compare the lengths of the three sides we found. We have $AB = \sqrt{37}$ units, $BC = \sqrt{37}$ units, and $CA = 2$ units. An isosceles triangle has at least two sides of equal length. Here, we see that $AB = BC$. Since two sides are equal, the triangle is isosceles.

Answer

Yes, the points $\mathbf{(5, -2)}$, $\mathbf{(6, 4)}$ and $\mathbf{(7, -2)}$ are the vertices of an isosceles triangle.