Question

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 YZ¯¯¯¯¯¯¯ ? Round your answer to one decimal place.(1 point) Side YZ¯¯¯¯¯¯¯ has a length of .

Answer 1

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

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

Here, the coordinates of the points are:

• \( Y(-4, 6) \)
• \( Z(-2, 2) \)

Using the distance formula, we can calculate the length of \( YZ \):

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

This simplifies to:

\[ YZ = \sqrt{(2)^2 + (-4)^2} \] \[ = \sqrt{4 + 16} \] \[ = \sqrt{20} \] \[ = \sqrt{4 \times 5} \] \[ = 2\sqrt{5} \]

Now, evaluating \( 2\sqrt{5} \approx 2 \times 2.236 = 4.472 \).

Rounding to one decimal place, the length of side \( YZ \) is approximately:

\[ \boxed{4.5} \]