Converse Parallelogram Proofs Practice

Question

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Question
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A parallelogram has vertices upper R 3m + 2, upper S 2n +2, upper U 8 + n, and upper T 3m - 3. Opposite vertices are connected via diagonal lines that intersect at an interior point labeled upper V. The lines formed between each vertex and upper V are labeled.

If quadrilateral RSTU
is a parallelogram, what are the lengths of the diagonals of the figure?

(1 point)
Diagonal RT
is equal to
and diagonal SU
is equal to
.
Remaining Attempts : 1

Answer 1

To find the lengths of the diagonals \( RT \) and \( SU \) in the parallelogram \( RSTU \), we can use the coordinates of the vertices given:

• \( R(3m + 2, y_R) \)
• \( S(2n + 2, y_S) \)
• \( U(8 + n, y_U) \)
• \( T(3m - 3, y_T) \)

In a parallelogram, the diagonals can be calculated using the distance formula, which is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. Calculate the length of diagonal \( RT \):

Using the points \( R(3m + 2, y_R) \) and \( T(3m - 3, y_T) \):

• Let's assume \( y_R \) and \( y_T \) are equal for the calculation, leading to:

\[ d_{RT} = \sqrt{( (3m - 3) - (3m + 2) )^2 + (y_T - y_R)^2} = \sqrt{(3m - 3 - 3m - 2)^2 + (y_T - y_R)^2} = \sqrt{(-5)^2 + 0^2} = \sqrt{25} = 5. \]

So, \( RT = 5 \).

2. Calculate the length of diagonal \( SU \):

Using the points \( S(2n + 2, y_S) \) and \( U(8 + n, y_U) \):

• Again assuming \( y_S \) and \( y_U \) are equal for the calculation:

\[ d_{SU} = \sqrt{( (8 + n) - (2n + 2) )^2 + (y_U - y_S)^2} = \sqrt{(8 + n - 2n - 2)^2 + (y_U - y_S)^2} = \sqrt{(6 - n)^2 + 0^2} = |6 - n|. \]

Thus, \( SU = |6 - n| \).

Final Answer:

• Diagonal \( RT \) is equal to \( 5 \)
• Diagonal \( SU \) is equal to \( |6 - n| \).