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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.One or more of your responses is incorrect.
At least one of your answers is incorrect. Identify a Pythagorean triple in which the greatest number is 17. The​ x-coordinate of Q is the​ x-coordinate of P plus the length of the horizontal leg. The​ y-coordinate of Q is the​ y-coordinate of P plus the length of the vertical leg.

1 answer

To solve the problem, we need to find integer coordinates for point \( Q \) given that \( P = (-4, 7) \) and the length of \( PQ \) is 17 units. We can use Pythagorean triples to achieve this.

Step 1: Identify a suitable Pythagorean triple

A Pythagorean triple consists of three positive integers \( a, b, c \) such that \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. We are looking for a triple where \( c = 17 \).

One Pythagorean triple that includes \( 17 \) is \( (8, 15, 17) \).

Step 2: Use the Pythagorean triple

Using the triple \( (8, 15, 17) \):

  • If we use \( 8 \) as the horizontal leg and \( 15 \) as the vertical leg, the coordinates of \( Q \) can be calculated as follows:

    • The x-coordinate of \( Q \) would be \( x_P + 8 = -4 + 8 = 4 \).
    • The y-coordinate of \( Q \) would be \( y_P + 15 = 7 + 15 = 22 \).

    Thus, one possible coordinate for \( Q \) is \( (4, 22) \).

  • Conversely, if we use \( 15 \) as the horizontal leg and \( 8 \) as the vertical leg, the coordinates of \( Q \) would be:

    • The x-coordinate of \( Q \) would be \( x_P + 15 = -4 + 15 = 11 \).
    • The y-coordinate of \( Q \) would be \( y_P + 8 = 7 + 8 = 15 \).

    Thus, another possible coordinate for \( Q \) is \( (11, 15) \).

Summary of Integer Coordinates

  1. Using the shorter leg horizontally:

    • \( Q = (4, 22) \)

  2. Using the shorter leg vertically:

    • \( Q = (11, 15) \)

Both coordinate pairs satisfy the condition that the x- and y-coordinates of \( Q \) are greater than those of \( P \) (i.e., both \( 4 > -4 \) and \( 22 > 7 \) for the first pair, and \( 11 > -4 \) and \( 15 > 7 \) for the second pair).