The locations of the vertices of quadrilateral LMNP are L(0, 0), M(2, 4), N(8, 6), and P(6, 2). Quadrilateral STUV is congruent to LMNP. What are the lengths of the diagonals of quadrilateral STUV?

To determine the lengths of the diagonals of quadrilateral \( STUV \), we first need to compute the lengths of the diagonals of quadrilateral \( LMNP \) since the two quadrilaterals are congruent. Congruent quadrilaterals have equal corresponding side lengths and diagonal lengths.

### Step 1: Identify the diagonals of \( LMNP \)

The diagonals of quadrilateral \( LMNP \) connect the following pairs of vertices:
- Diagonal 1: \( L(0, 0) \) to \( N(8, 6) \)
- Diagonal 2: \( M(2, 4) \) to \( P(6, 2) \)

### Step 2: Compute the lengths of the diagonals

To calculate the lengths of the diagonals, we use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

#### Diagonal 1: \( L(0, 0) \) to \( N(8, 6) \)
\[
d_{LN} = \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

#### Diagonal 2: \( M(2, 4) \) to \( P(6, 2) \)
\[
d_{MP} = \sqrt{(6 - 2)^2 + (2 - 4)^2} = \sqrt{(4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.47
\]

### Step 3: Conclusion

Since quadrilateral \( STUV \) is congruent to quadrilateral \( LMNP \), the lengths of the diagonals of \( STUV \) are the same as those of \( LMNP \):
- Length of Diagonal 1: \( 10 \)
- Length of Diagonal 2: \( 4.47 \)

Do you want further details or have any questions?

Here are five related questions to extend this topic:
1. How would you compute the perimeter of quadrilateral \( LMNP \)?
2. What is the area of quadrilateral \( LMNP \)?
3. How do you prove two quadrilaterals are congruent?
4. How can you find the coordinates of quadrilateral \( STUV \) if given a transformation?
5. How do you find the angles within a quadrilateral based on its vertex coordinates?

**Tip:** When using the distance formula, double-check the signs of the coordinates to avoid calculation errors.

Learn how to calculate the lengths of the diagonals of quadrilateral LMNP using the distance formula and congruence. This geometry problem is ideal for students in Grades 8-10, focusing on key concepts like congruent quadrilaterals and the distance formula.

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