Amy is trying to prove that a triangle with the vertices X (-6, 2), Y (-4, 6), and Z (-2, 2) is an isosceles triangle. She found that side XY has a length of approximately 4.5 and that side XZ has a length of 4. What is the length of side Y Z? Round your answer to one decimal place.

1 answer

To find the length of side \( YZ \), we will use the distance formula, which is given by:

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

The coordinates of points \( Y \) and \( Z \) are:

  • \( Y (-4, 6) \)
  • \( Z (-2, 2) \)

Now we can apply the distance formula to find \( YZ \):

\[ YZ = \sqrt{((-2) - (-4))^2 + (2 - 6)^2} \]

Calculating the differences:

\[ (-2) - (-4) = -2 + 4 = 2 \] \[ 2 - 6 = -4 \]

Substituting these values into the formula gives:

\[ YZ = \sqrt{(2)^2 + (-4)^2} \]

Calculating the squares:

\[ YZ = \sqrt{4 + 16} = \sqrt{20} \]

Simplifying \( \sqrt{20} \):

\[ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} \]

Now, we can calculate numerical value for \( 2\sqrt{5} \):

Using \( \sqrt{5} \approx 2.236 \):

\[ 2\sqrt{5} \approx 2 \cdot 2.236 \approx 4.472 \]

Rounding to one decimal place:

\[ 4.472 \approx 4.5 \]

So, the length of side \( YZ \) is approximately \( \boxed{4.5} \).