The coordinates of the vertices of quadrilateral RSTU are R(−4, 1), S(4, −1), T(3, −6), and U(−5, −4).

 

Which statement correctly describes whether quadrilateral RSTU is a rectangle?
Responses

Quadrilateral RSTU is not a rectangle because it has no right angles.
quadrilateral R S T U is not a rectangle because it has no right angles.

Quadrilateral RSTU is not a rectangle because it has only two right angles.
quadrilateral R S T U is not a rectangle because it has only two right angles.

Quadrilateral RSTU is not a rectangle because it has only one right angle.
quadrilateral R S T U is not a rectangle because it has only one right angle.

Quadrilateral RSTU is a rectangle because it has four right angles.

To determine whether quadrilateral RSTU is a rectangle, we need to check if all four angles of the quadrilateral are right angles. For this, we will use the concept that two lines are perpendicular (and thus form a right angle) if the product of their slopes is \(-1\).

### Step 1: Find the slopes of the sides of the quadrilateral

- **Slope of RS** (between R(-4, 1) and S(4, -1)):
  \[
  \text{Slope of RS} = \frac{-1 - 1}{4 - (-4)} = \frac{-2}{8} = -\frac{1}{4}
  \]

- **Slope of ST** (between S(4, -1) and T(3, -6)):
  \[
  \text{Slope of ST} = \frac{-6 - (-1)}{3 - 4} = \frac{-5}{-1} = 5
  \]

- **Slope of TU** (between T(3, -6) and U(-5, -4)):
  \[
  \text{Slope of TU} = \frac{-4 - (-6)}{-5 - 3} = \frac{2}{-8} = -\frac{1}{4}
  \]

- **Slope of UR** (between U(-5, -4) and R(-4, 1)):
  \[
  \text{Slope of UR} = \frac{1 - (-4)}{-4 - (-5)} = \frac{5}{1} = 5
  \]

### Step 2: Check for perpendicularity

- **RS and ST**: The product of their slopes is \(-\frac{1}{4} \times 5 = -\frac{5}{4}\) (not \(-1\)).
- **ST and TU**: The product of their slopes is \(5 \times -\frac{1}{4} = -\frac{5}{4}\) (not \(-1\)).
- **TU and UR**: The product of their slopes is \(-\frac{1}{4} \times 5 = -\frac{5}{4}\) (not \(-1\)).
- **UR and RS**: The product of their slopes is \(5 \times -\frac{1}{4} = -\frac{5}{4}\) (not \(-1\)).

### Conclusion
None of the consecutive sides of quadrilateral RSTU are perpendicular to each other, so there are no right angles in the quadrilateral. 

**Correct Statement:** Quadrilateral RSTU is not a rectangle because it has no right angles.

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Would you like more details or have any questions?

Here are some related questions:
1. How do you find the area of a quadrilateral using its vertices?
2. What is the definition of a parallelogram, and how does it differ from a rectangle?
3. How can the distance formula be used to verify if a quadrilateral is a square?
4. What are the properties of different types of quadrilaterals?
5. How do you use the slope formula to determine if two lines are parallel?

**Tip:** Always double-check your slope calculations, as they are crucial in determining the properties of geometric shapes.

Explore whether quadrilateral RSTU is a rectangle based on the coordinates of its vertices. Understand the concept of right angles in quadrilaterals and how to use slope calculations to determine perpendicularity. Suitable for high school students studying geometry.

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