To find the lengths of the sides \( \overline{AB} \) and \( \overline{BC} \) in triangle ABC and hence the length \( \overline{PQ} \) in triangle PQR needed to use the HL (Hypotenuse-Leg) Congruence Theorem, you would first need the coordinates of the points A, B, C, P, Q, and R.
Assuming you have the coordinates, you can use the distance formula to find the lengths of the sides:
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculate Length of \( \overline{AB} \):
- If point A is at \( (x_1, y_1) \) and point B at \( (x_2, y_2) \): \[ \overline{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculate Length of \( \overline{BC} \):
- If point B is at \( (x_2, y_2) \) and point C at \( (x_3, y_3) \): \[ \overline{BC} = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \]
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Determine Length of \( \overline{PQ} \):
- To satisfy the HL Theorem, you need the lengths of both hypotenuses equal. Assume the hypotenuse of triangle ABC (between A and C) is \( \overline{AC} \). The same applies to triangle PQR where you need to find hypotenuse \( \overline{PQ} \) that should be equal to \( \overline{AC} \).
- Calculate \( \overline{AC} \) as: \[ \overline{AC} = \sqrt{(x_3 - x_1)^2 + (y_3 - y_1)^2} \]
Without the specific coordinates, I can't provide exact numerical values for \( \overline{AB} \), \( \overline{BC} \), and \( \overline{PQ} \). Please provide the coordinates for the points or calculate using the process outlined above.
Once you have the lengths determined, you can fill in the blanks for:
- \( ab = \text{length of } \overline{AB} \)
- \( bc = \text{length of } \overline{BC} \)
- \( pq = \text{length of } \overline{PQ} \) which equals \( \overline{AC} \) to satisfy HL congruence.