Question

triangle xyz is shown on the coordinate grid. which statements are true about triangle xyz? choose three correct answers. xyz is a scalene triangle. xy measures \\(\sqrt{26}\\) units. xyz is a right triangle.

Explanation:

To solve this, we first find the lengths of the sides of triangle \( XYZ \) using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step 1: Find length of \( XY \)

Points \( X(-1, 5) \) and \( Y(4, 4) \).
\[

$$\begin{align*} XY &= \sqrt{(4 - (-1))^2 + (4 - 5)^2}\\ &= \sqrt{(5)^2 + (-1)^2}\\ &= \sqrt{25 + 1}\\ &= \sqrt{26} \end{align*}$$

\]

Step 2: Find length of \( YZ \)

Points \( Y(4, 4) \) and \( Z(-2, 0) \).
\[

$$\begin{align*} YZ &= \sqrt{(-2 - 4)^2 + (0 - 4)^2}\\ &= \sqrt{(-6)^2 + (-4)^2}\\ &= \sqrt{36 + 16}\\ &= \sqrt{52} = 2\sqrt{13} \end{align*}$$

\]

Step 3: Find length of \( XZ \)

Points \( X(-1, 5) \) and \( Z(-2, 0) \).
\[

$$\begin{align*} XZ &= \sqrt{(-2 - (-1))^2 + (0 - 5)^2}\\ &= \sqrt{(-1)^2 + (-5)^2}\\ &= \sqrt{1 + 25}\\ &= \sqrt{26} \end{align*}$$

\]

Step 4: Analyze the triangle

• Scalene? No, because \( XY = XZ = \sqrt{26} \), so it's isoceles, not scalene.
• \( XY \) length? Yes, \( XY = \sqrt{26} \) (from Step 1).
• Right triangle? Check using Pythagorean theorem. Let's see if \( XY^2 + XZ^2 = YZ^2 \):

\[

$$\begin{align*} (\sqrt{26})^2 + (\sqrt{26})^2 &= 26 + 26 = 52\\ (2\sqrt{13})^2 &= 4 \times 13 = 52 \end{align*}$$

\]
So \( XY^2 + XZ^2 = YZ^2 \), so it is a right triangle (right-angled at \( X \)).

Assuming the other options (not fully visible) are checked similarly, but from the visible ones:

• "XYZ is a scalene triangle." → False.
• "XY measures \( \sqrt{26} \) units." → True (from Step 1).
• "XYZ is a right triangle." → True (from Step 4).

For the three correct answers (assuming the third is another true statement, e.g., if there's an option like "XZ measures \( \sqrt{26} \) units" or similar, but from the visible and our calculation):

1. \( \boldsymbol{\text{XY measures } \sqrt{26} \text{ units}} \)
2. \( \boldsymbol{\text{XYZ is a right triangle}} \)
3. (Another true one, e.g., "XZ measures \( \sqrt{26} \) units" if present)

If we consider the visible options and our analysis, the two checked (and the third correct one) are:

• \( \boldsymbol{\text{XY measures } \sqrt{26} \text{ units}} \)
• \( \boldsymbol{\text{XYZ is a right triangle}} \)
• (And one more, like "XZ measures \( \sqrt{26} \) units" if that's an option)

Final Answer (for the two visible correct and one more):
The three correct statements are (example with common true ones):

• \( \boldsymbol{\text{XY measures } \sqrt{26} \text{ units}} \)
• \( \boldsymbol{\text{XYZ is a right triangle}} \)
• \( \boldsymbol{\text{XZ measures } \sqrt{26} \text{ units}} \) (or another true statement based on full options)

Answer:

To solve this, we first find the lengths of the sides of triangle \( XYZ \) using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step 1: Find length of \( XY \)

Points \( X(-1, 5) \) and \( Y(4, 4) \).
\[

$$\begin{align*} XY &= \sqrt{(4 - (-1))^2 + (4 - 5)^2}\\ &= \sqrt{(5)^2 + (-1)^2}\\ &= \sqrt{25 + 1}\\ &= \sqrt{26} \end{align*}$$

\]

Step 2: Find length of \( YZ \)

Points \( Y(4, 4) \) and \( Z(-2, 0) \).
\[

$$\begin{align*} YZ &= \sqrt{(-2 - 4)^2 + (0 - 4)^2}\\ &= \sqrt{(-6)^2 + (-4)^2}\\ &= \sqrt{36 + 16}\\ &= \sqrt{52} = 2\sqrt{13} \end{align*}$$

\]

Step 3: Find length of \( XZ \)

Points \( X(-1, 5) \) and \( Z(-2, 0) \).
\[

$$\begin{align*} XZ &= \sqrt{(-2 - (-1))^2 + (0 - 5)^2}\\ &= \sqrt{(-1)^2 + (-5)^2}\\ &= \sqrt{1 + 25}\\ &= \sqrt{26} \end{align*}$$

\]

Step 4: Analyze the triangle

• Scalene? No, because \( XY = XZ = \sqrt{26} \), so it's isoceles, not scalene.
• \( XY \) length? Yes, \( XY = \sqrt{26} \) (from Step 1).
• Right triangle? Check using Pythagorean theorem. Let's see if \( XY^2 + XZ^2 = YZ^2 \):

\[

$$\begin{align*} (\sqrt{26})^2 + (\sqrt{26})^2 &= 26 + 26 = 52\\ (2\sqrt{13})^2 &= 4 \times 13 = 52 \end{align*}$$

\]
So \( XY^2 + XZ^2 = YZ^2 \), so it is a right triangle (right-angled at \( X \)).

Assuming the other options (not fully visible) are checked similarly, but from the visible ones:

• "XYZ is a scalene triangle." → False.
• "XY measures \( \sqrt{26} \) units." → True (from Step 1).
• "XYZ is a right triangle." → True (from Step 4).

For the three correct answers (assuming the third is another true statement, e.g., if there's an option like "XZ measures \( \sqrt{26} \) units" or similar, but from the visible and our calculation):

1. \( \boldsymbol{\text{XY measures } \sqrt{26} \text{ units}} \)
2. \( \boldsymbol{\text{XYZ is a right triangle}} \)
3. (Another true one, e.g., "XZ measures \( \sqrt{26} \) units" if present)

If we consider the visible options and our analysis, the two checked (and the third correct one) are:

• \( \boldsymbol{\text{XY measures } \sqrt{26} \text{ units}} \)
• \( \boldsymbol{\text{XYZ is a right triangle}} \)
• (And one more, like "XZ measures \( \sqrt{26} \) units" if that's an option)

Final Answer (for the two visible correct and one more):
The three correct statements are (example with common true ones):

• \( \boldsymbol{\text{XY measures } \sqrt{26} \text{ units}} \)
• \( \boldsymbol{\text{XYZ is a right triangle}} \)
• \( \boldsymbol{\text{XZ measures } \sqrt{26} \text{ units}} \) (or another true statement based on full options)