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

Answer 1

To determine the length of side \( YZ \) for the triangle with vertices \( X(-6,2) \), \( Y(-4,6) \), and \( Z(-2,2) \), we can use the distance formula, which is given by:

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

First, let’s find the length of side \( YZ \):

1. **Coordinates of points \( Y \) and \( Z \):**
- \( Y(-4, 6) \)
- \( Z(-2, 2) \)

2. **Applying the distance formula:**
\[
YZ = \sqrt{((-2) - (-4))^2 + (2 - 6)^2}
\]
\[
YZ = \sqrt{(-2 + 4)^2 + (2 - 6)^2}
\]
\[
YZ = \sqrt{(2)^2 + (-4)^2}
\]
\[
YZ = \sqrt{4 + 16}
\]
\[
YZ = \sqrt{20}
\]
\[
YZ = 2\sqrt{5}
\]

3. **Calculating \( 2\sqrt{5} \):**
- First, calculate \( \sqrt{5} \):
\[
\sqrt{5} \approx 2.236
\]
- Then calculate \( 2\sqrt{5} \):
\[
2\sqrt{5} \approx 2 \times 2.236 \approx 4.472
\]

4. **Rounding to one decimal place:**
\[
YZ \approx 4.5
\]

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