Math Problem Statement

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?

Solution

To determine the lengths of the diagonals of quadrilateral STUVSTUV, we first need to compute the lengths of the diagonals of quadrilateral LMNPLMNP since the two quadrilaterals are congruent. Congruent quadrilaterals have equal corresponding side lengths and diagonal lengths.

Step 1: Identify the diagonals of LMNPLMNP

The diagonals of quadrilateral LMNPLMNP connect the following pairs of vertices:

  • Diagonal 1: L(0,0)L(0, 0) to N(8,6)N(8, 6)
  • Diagonal 2: M(2,4)M(2, 4) to P(6,2)P(6, 2)

Step 2: Compute the lengths of the diagonals

To calculate the lengths of the diagonals, we use the distance formula: d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points.

Diagonal 1: L(0,0)L(0, 0) to N(8,6)N(8, 6)

dLN=(80)2+(60)2=82+62=64+36=100=10d_{LN} = \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

Diagonal 2: M(2,4)M(2, 4) to P(6,2)P(6, 2)

dMP=(62)2+(24)2=(4)2+(2)2=16+4=204.47d_{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 STUVSTUV is congruent to quadrilateral LMNPLMNP, the lengths of the diagonals of STUVSTUV are the same as those of LMNPLMNP:

  • Length of Diagonal 1: 1010
  • Length of Diagonal 2: 4.474.47

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 LMNPLMNP?
  2. What is the area of quadrilateral LMNPLMNP?
  3. How do you prove two quadrilaterals are congruent?
  4. How can you find the coordinates of quadrilateral STUVSTUV 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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Congruence
Distance Formula
Quadrilaterals

Formulas

Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Congruent figures have equal corresponding side lengths and diagonals

Suitable Grade Level

Grades 8-10