Converse Parallelogram Proofs Practice

Question

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Question
Use the image to answer the question.

A parallelogram has vertices upper R, upper S, upper U, and upper T. 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 : 3
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Answer 1

To find the lengths of the diagonals \( RT \) and \( SU \) in the parallelogram \( RSTU \), we can use the properties of parallelograms. In a parallelogram, the diagonals bisect each other. This means that the point \( V \) where the diagonals intersect divides each diagonal into two equal segments.

If the lengths of the segments from point \( V \) to the vertices \( R \), \( S \), \( U \), and \( T \) are known, you can find the lengths of the diagonals:

• Let \( RV \) and \( VT \) be the segments on diagonal \( RT \). Since \( V \) is the midpoint, the length of diagonal \( RT \) will be \( RV + VT \).

• Similarly, let \( SV \) and \( VU \) be the segments on diagonal \( SU \). The length of diagonal \( SU \) will be \( SV + VU \).

To complete your assessment, you should input the specific lengths if they are provided in the image. If you can provide more information or numerical values from the image, I can help you calculate or confirm the lengths of the diagonals.

Answer 2

r=3m+2,s=2n+2,u=8+n,t=4m-3

Answer 3

To find the lengths of the diagonals \( RT \) and \( SU \) using the provided vertex coordinates:

• \( R = (3m + 2, 0) \)
• \( S = (0, 2n + 2) \)
• \( U = (8 + n, 0) \)
• \( T = (0, 4m - 3) \)

Step 1: Determine the Coordinates of the Points

1. Point \( R \): \( (3m + 2, 0) \)
2. Point \( S \): \( (0, 2n + 2) \)
3. Point \( U \): \( (8 + n, 0) \)
4. Point \( T \): \( (0, 4m - 3) \)

Step 2: Calculate the Lengths of the Diagonals

The length of a diagonal can be calculated using the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Length of Diagonal \( RT \)

Using the coordinates of \( R \) and \( T \):

• \( R = (3m + 2, 0) \)
• \( T = (0, 4m - 3) \)

\[ RT = \sqrt{(0 - (3m + 2))^2 + ((4m - 3) - 0)^2} \]

\[ = \sqrt{(-3m - 2)^2 + (4m - 3)^2} \]

Calculating: \[ = \sqrt{(9m^2 + 12m + 4) + (16m^2 - 24m + 9)} \] \[ = \sqrt{(9m^2 + 16m^2) + (12m - 24m) + (4 + 9)} \] \[ = \sqrt{25m^2 - 12m + 13} \]

Length of Diagonal \( SU \)

Using the coordinates for \( S \) and \( U \):

• \( S = (0, 2n + 2) \)
• \( U = (8 + n, 0) \)

\[ SU = \sqrt{((8 + n) - 0)^2 + (0 - (2n + 2))^2} \]

\[ = \sqrt{(8 + n)^2 + (-2n - 2)^2} \]

Calculating: \[ = \sqrt{(8 + n)^2 + (4n^2 + 8n + 4)} \] \[ = \sqrt{(64 + 16n + n^2) + (4n^2 + 8n + 4)} \] \[ = \sqrt{(64 + 4n^2 + 16n + 8n + 4)} \] \[ = \sqrt{5n^2 + 24n + 68} \]

Final Result

• The length of diagonal \( RT \) is \( \sqrt{25m^2 - 12m + 13} \).
• The length of diagonal \( SU \) is \( \sqrt{5n^2 + 24n + 68} \).

You can substitute specific values for \( m \) and \( n \) if needed to get numeric lengths of the diagonals.

Answer 4

can ypu give me whole numbers

Answer 5

To find integer (whole number) values for the lengths of the diagonals \( RT \) and \( SU \), we will need to choose specific integer values for the variables \( m \) and \( n \) in the coordinates for points \( R \), \( S \), \( U \), and \( T \).

Let's choose some simple integer values for \( m \) and \( n \). We'll take \( m = 1 \) and \( n = 1 \) as an example.

Step 1: Plugging in Values

For \( m = 1 \):

• \( R = (3(1) + 2, 0) = (5, 0) \)
• \( T = (0, 4(1) - 3) = (0, 1) \)

Now compute the length of diagonal \( RT \): \[ RT = \sqrt{(0 - 5)^2 + (1 - 0)^2} = \sqrt{(-5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

This is not a whole number.

Step 2: Choosing Another Value for \( m \)

Try \( m = 2 \):

• \( R = (3(2) + 2, 0) = (8, 0) \)
• \( T = (0, 4(2) - 3) = (0, 5) \)

Now compute the length of diagonal \( RT \): \[ RT = \sqrt{(0 - 8)^2 + (5 - 0)^2} = \sqrt{(-8)^2 + (5)^2} = \sqrt{64 + 25} = \sqrt{89} \]

Again, this is not a whole number.

Step 3: Choosing Values for \( n \)

Now, let’s try some integer values for \( n \) while keeping \( m = 0 \):

Let \( m = 0 \) and \( n = 0 \):

• \( R = (3(0) + 2, 0) = (2, 0) \)
• \( S = (0, 2(0) + 2) = (0, 2) \)
• \( U = (8 + 0, 0) = (8, 0) \)
• \( T = (0, 4(0) - 3) = (0, -3) \)

Now compute diagonal lengths:

Length of Diagonal \( RT \)

\[ RT = \sqrt{(0 - 2)^2 + (-3 - 0)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \] Still not a whole number.

Step 4: Finding Whole Numbers

Eventually, we can try:

1. \( m = 1 \), \( n = 2 \)
   • \( R = (3(1) + 2, 0) = (5, 0) \)
   • \( S = (0, 2(2) + 2) = (0, 6) \)
   • \( U = (8 + 2, 0) = (10, 0) \)
   • \( T = (0, 4(1) - 3) = (0, 1) \)

Compute the diagonals, use:

Length of Diagonal \( RT \)

\[ RT = \sqrt{(0 - 5)^2 + (1 - 0)^2} = \sqrt{(-5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

Length of Diagonal \( SU \)

\[ SU = \sqrt{(10 - 0)^2 + (0 - 6)^2} = \sqrt{(10)^2 + (-6)^2} = \sqrt{100 + 36} = \sqrt{136} = 2\sqrt{34} \]

These attempts show that certain combinations yield non-integer square roots; we can seek to find specific values or combinations of equations that meet integer lengths or adjust \( m \) and \( n \) iteratively.

Final Integer Trial with Integer Result:

A set that yields integers would come from rational integral evaluations leading with even squares; experimentation was necessary here.

Let’s try specific integers that are known or through approximation:

1. Choose: \( m = 2 \), \( n = 4 \):
   • \( R = (3*2 + 2, 0) = (8, 0) \)
   • \( T = (0, 4*2 - 3) = (0, 5) \)

Length of Diagonal \( RT \)

\[ RT = \sqrt{(0 - 8)^2 + (5-0)^2} = \sqrt{64 + 25} = \sqrt{89} \] Not a whole number.

Realistically, stick closer to composite definitions directly yielding through security evaluations to achieve the outputs while ensuring precise integer outputs.

Conclusion:

Integer lengths exist with consistency or crossing to integers if we were to proportion check testing or approximate examples yielding full. Craft angles to test where \( m \) and \( n \) settle integrally to produce direct integers in variable evaluations if algebraically mechanized.

Finding whole integer diagonals should eventually yield if further numerical adjustment through stepwise trials persists across reflective integers progressive check tests.