Bot GPT 4-o mini To determine the values of \( x \) and \( y \) for which quadrilateral ABCD is a parallelogram, we need to use the property that in a parallelogram, opposite sides are equal in length.
We have the following side lengths for quadrilateral ABCD:
- \( AB = 20y \)
- \( BC = 1 + 3x \)
- \( CD = 4y + 32 \)
- \( AD = 6x - 20 \)
For quadrilateral ABCD to be a parallelogram, the following equalities must hold:
- \( AB = CD \)
- \( BC = AD \)
Now, let's set up the equations based on these equalities:
1. From \( AB = CD \): \[ 20y = 4y + 32 \]
To solve for \( y \), we can isolate \( y \): \[ 20y - 4y = 32 \] \[ 16y = 32 \] \[ y = \frac{32}{16} = 2 \]
2. From \( BC = AD \): \[ 1 + 3x = 6x - 20 \]
Now, let's solve for \( x \): \[ 1 + 20 = 6x - 3x \] \[ 21 = 3x \] \[ x = \frac{21}{3} = 7 \]
Final Values: The values of \( x \) and \( y \) for which quadrilateral ABCD is a parallelogram are: \[ x = 7, \quad y = 2 \]
Thus, Jonah can be sure that his quadrilateral \( ABCD \) is a parallelogram when \( x = 7 \) and \( y = 2 \).
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