Question

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Part 1
PQ overbar has a length of 17 units with ​P(minus​4,7). If the​ x- and​ y-coordinates of Q are both greater than the​ x- and​ y-coordinates of​ P, what are possible integer value coordinates of​ Q? Explain.
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Part 1
Let PQ overbar be the hypotenuse of a right triangle that also has a horizontal leg and a vertical leg. The hypotenuse then has length​ 17, and a Pythagorean triple can be used to say the shorter leg has length
  
enter your response here and the longer leg has length
  
enter your response here. If the shorter leg is​ horizontal, then Q is described by the ordered pair
  
enter your response here. If the shorter leg is​ vertical, then Q is described by the ordered pair
  
enter your response here.

Answer 1

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:

1. \( (x - (-4))^2 + (y - 7)^2 = 17^2 \)
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:

1. 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:

1. \( 8^2 + 15^2 = 64 + 225 = 289 \)
2. \( 15^2 + 8^2 = 225 + 64 = 289 \)
3. \( 0^2 + 17^2 = 0 + 289 = 289 \) (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:

1. **If the shorter leg is horizontal, then**:
- Q: (11, 15)

2. **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: \( (4, 22) \)