Question

Coordinate Proofs Practice Complete this assessment to review what you’ve learned. It will not count toward your grade. 2 of 52 of 5 Items 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 . Remaining Attempts : 3 Skip to navigation page 17 of 18 Formula keypad has been closed. Press Control + Backslash to open it again.

Answer 1

To find the length of side \( YZ \) in triangle \( XYZ \) with vertices \( X(-6, 2) \), \( Y(-4, 6) \), and \( Z(-2, 2) \), we can use the distance formula:

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

Let's identify the points:
- \( Y(-4, 6) \)
- \( Z(-2, 2) \)

Now we can calculate the length of \( YZ \):

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

Calculating the differences:

\[
= \sqrt{(-2 + 4)^2 + (2 - 6)^2}
\]
\[
= \sqrt{(2)^2 + (-4)^2}
\]
\[
= \sqrt{4 + 16}
\]
\[
= \sqrt{20}
\]
\[
= 2\sqrt{5}
\]

Now, approximating \( 2\sqrt{5} \):

First, we know that \( \sqrt{5} \approx 2.236 \), so:

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

Rounding to one decimal place:

\[
\text{Length of } YZ \approx 4.5
\]

Thus, side \( YZ \) has a length of approximately **4.5**.