Question

the map of a farmers property is displayed on a coordinate plane. there are two silos on the property: silo a and silo b. complete the parts below to find the distance between the two silos. (a) on the map, silo a is at (-3, -5) and the silo b is at (2, -1). draw a right triangle so that the hypotenuse is the segment between these two points. make sure to draw all three sides of the triangle using the button. (b) let x be the length of the hypotenuse. use the triangle you drew and the pythagorean theorem to fill in the blanks. \\(\square^2 + \square^2 = x^2\\)

Explanation:

Part (a) Explanation (Drawing the Right Triangle):

To draw the right triangle with hypotenuse between \( (-3, -5) \) (Silo A) and \( (2, -1) \) (Silo B):

1. Horizontal Leg: Move horizontally from Silo A to the point directly below Silo B (or vice versa). The \( x \)-coordinates are \( -3 \) and \( 2 \), so the horizontal distance is \( |2 - (-3)| = 5 \) units. This leg is parallel to the \( x \)-axis.
2. Vertical Leg: Move vertically from the horizontal endpoint to Silo B. The \( y \)-coordinates are \( -5 \) and \( -1 \), so the vertical distance is \( |-1 - (-5)| = 4 \) units. This leg is parallel to the \( y \)-axis.
3. Hypotenuse: Connect Silo A to Silo B (the hypotenuse).

Part (b) Explanation (Pythagorean Theorem):

The Pythagorean Theorem states \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the right triangle, and \( c \) is the hypotenuse (\( x \) here).

• The horizontal leg (along the \( x \)-axis) has length \( 5 \) (from \( x = -3 \) to \( x = 2 \), so \( 2 - (-3) = 5 \)).
• The vertical leg (along the \( y \)-axis) has length \( 4 \) (from \( y = -5 \) to \( y = -1 \), so \( -1 - (-5) = 4 \)).

Step-by-Step for Part (b):

Step 1: Identify Leg Lengths

Horizontal leg length: \( |2 - (-3)| = 5 \)
Vertical leg length: \( |-1 - (-5)| = 4 \)

Step 2: Apply Pythagorean Theorem

Substitute \( a = 5 \), \( b = 4 \), and \( c = x \) into \( a^2 + b^2 = x^2 \):
$$ 5^2 + 4^2 = x^2 $$

Answer:

(Part b):
The blanks should be filled with \( 5 \) and \( 4 \) (order does not matter). So:
\( \boldsymbol{5}^2 + \boldsymbol{4}^2 = x^2 \)