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 find the lengths of the diagonals of quadrilateral \(STUV\), we must first determine the lengths of the diagonals of quadrilateral \(LMNP\), since \(STUV\) is congruent to \(LMNP\) and congruent shapes have equal corresponding sides and diagonals.

### Diagonals of Quadrilateral LMNP:

1. **Diagonal \(LN\):**
   The coordinates of \(L\) and \(N\) are \(L(0, 0)\) and \(N(8, 6)\), respectively.
   To find the length of diagonal \(LN\), use the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting the coordinates of \(L\) and \(N\):
   \[
   LN = \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10
   \]

2. **Diagonal \(MP\):**
   The coordinates of \(M\) and \(P\) are \(M(2, 4)\) and \(P(6, 2)\), respectively.
   Using the distance formula:
   \[
   MP = \sqrt{(6 - 2)^2 + (2 - 4)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
   \]

### Conclusion:
Since quadrilateral \(STUV\) is congruent to \(LMNP\), the lengths of its diagonals will be the same as those of \(LMNP\):
- Length of diagonal \(SU = LN = 10\)
- Length of diagonal \(TV = MP = 2\sqrt{5}\)

Would you like more details or have any questions?

### Related Questions:
1. How do you prove two quadrilaterals are congruent?
2. What are other properties of congruent quadrilaterals?
3. How can you find the area of a quadrilateral using its vertices?
4. How would the result change if quadrilateral \(STUV\) were not congruent to \(LMNP\)?
5. Can you find the perimeter of quadrilateral \(LMNP\) using the vertex coordinates?

### Tip:
Always use the distance formula carefully by double-checking coordinate differences, as a small mistake can change the outcome.

Explore how to find the lengths of the diagonals of quadrilateral LMNP using the distance formula and congruence properties. This detailed guide helps you solve similar problems with congruent quadrilaterals, ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation