To find the possible integer coordinates for point Q when the line segment PQ has a length of 17 units and point P is located at (-4, 7), we can utilize the Pythagorean theorem. Since the coordinates of Q must be greater than those of P, we will find the integer solutions (x, y) such that:
- \( (x - (-4))^2 + (y - 7)^2 = 17^2 \)
- The coordinates of Q (x, y) must be greater than (-4, 7); therefore, x > -4 and y > 7.
Step 1: Set up the equation
Using the distance formula:
\[ (x + 4)^2 + (y - 7)^2 = 17^2 \] \[ (x + 4)^2 + (y - 7)^2 = 289 \]
Step 2: Find Pythagorean triples
Next, we can find integer solutions where:
- Define the horizontal leg as \( a \) and the vertical leg as \( b \): \[ a^2 + b^2 = 289 \]
Step 3: Find possible pairs (a, b)
By factoring 289, we can find the Pythagorean triples. The perfect integer pairs \( (a, b) \) can be:
- \( 8^2 + 15^2 = 64 + 225 = 289 \)
- \( 15^2 + 8^2 = 225 + 64 = 289 \)
- \( 0^2 + 17^2 = 0 + 289 = 289 \) (not useful since coordinates of Q must be greater than P).
Thus, we derive the following combinations:
- Horizontal leg = 15, vertical leg = 8
- Horizontal leg = 8, vertical leg = 15
Step 4: Determine coordinates of Q
Case 1: Shorter leg is horizontal (15) and longer leg is vertical (8)
- If the horizontal distance (to the right of P) is 15:
- \( x = -4 + 15 = 11 \)
- \( y = 7 + 8 = 15 \)
- Q = (11, 15)
Case 2: Shorter leg is vertical (8) and longer leg is horizontal (15)
- If the horizontal distance (to the right of P) is 8:
- \( x = -4 + 8 = 4 \)
- \( y = 7 + 15 = 22 \)
- Q = (4, 22)
Conclusion
Thus, the possible integer coordinates for Q with respect to point P as (-4, 7) and considering the conditions given are:
-
If the shorter leg is horizontal, then:
- Q: (11, 15)
-
If the shorter leg is vertical, then:
- Q: (4, 22)
So, the final answers would be:
- \( \text{Shorter leg length} = 8 \), \( \text{Longer leg length} = 15 \)
- Q for shorter horizontal leg: \( (11, 15) \)
- Q for shorter vertical leg: \( (4, 22) \)
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