Question

Joaquin drew the triangle below.

On a coordinate plane, triangle K L J has points (3, 6), (4, 0) and (negative 5, 0).

Which statement must be true about a figure that is congruent to Joaquin’s triangle?
It has two angles on the x-axis.
It has a side that is 9 units long.
It has a side that lies on the x-axis.
It has an obtuse angle.

Answer 1

To analyze the triangle K L J with points \( K(3, 6) \), \( L(4, 0) \), and \( J(-5, 0) \), let's first consider the properties of this triangle.

1. **Coordinates of Points**:
- Point \( L(4, 0) \) and point \( J(-5, 0) \) are both located on the x-axis.
- Point \( K(3, 6) \) is above the x-axis.

2. **Side Lengths**:
- The length of side \( LJ \) can be calculated using the distance formula:
\[
LJ = |4 - (-5)| = |4 + 5| = 9 \text{ units}
\]
- The length of side \( KL \) and side \( KJ \) would need to be calculated for a complete understanding, but it's already evident that \( LJ \) is 9 units long.

3. **Angles**:
- Since \( L \) and \( J \) are on the x-axis and point \( K \) is above the x-axis, triangle \( K L J \) likely contains at least one angle greater than 90 degrees, as \( K \) is higher than both \( L \) and \( J \).

Given these observations, let's evaluate each option:

- **It has two angles on the x-axis**: Not necessarily true; while \( L \) and \( J \) are on the x-axis, it doesn't mean any congruent triangle has to have angles at those points.

- **It has a side that is 9 units long**: This must be true since \( LJ \) is 9 units long. Any triangle congruent to triangle \( K L J \) will also have a side that is 9 units long.

- **It has a side that lies on the x-axis**: \( LJ \) lies on the x-axis, and since congruent triangles maintain the same angles and lengths, a congruent triangle must also have a side on the x-axis.

- **It has an obtuse angle**: While triangle \( K L J \) is likely to contain obtuse angles, congruent triangles can have various orientations and may not necessarily retain this characteristic.

Based on this analysis, the statements that **must be true** about a triangle congruent to Joaquin’s triangle are:

- It has a side that is 9 units long.
- It has a side that lies on the x-axis.

Among these options, **the strongest statement that is definitively true is**:

**It has a side that is 9 units long.**