Question

question 1 multiple choice (worth 1 points) (04.03 mc) alex wants to fence in an area for a dog park. he has plotted three sides of the fenced area at the points e (1, 5), f (3, 5), and g (6, 1). he has 16 units of fencing. where could alex place point h so that he does not have to buy more fencing? options: (0, 1), (0, -2), (1, 1), (1, -2)

Explanation:

Answer:

First, find the lengths of the existing sides:

• \( E(1,5) \) to \( F(3,5) \): horizontal distance, length \( 3 - 1 = 2 \).
• \( F(3,5) \) to \( G(6,1) \): use distance formula \( \sqrt{(6 - 3)^2 + (1 - 5)^2} = \sqrt{9 + 16} = 5 \).
• \( E(1,5) \) to \( G(6,1) \): \( \sqrt{(6 - 1)^2 + (1 - 5)^2} = \sqrt{25 + 16} = \sqrt{41} \) (not needed here).

Total fencing used for three sides: \( 2 + 5 + \text{length of } EH \text{ or } GH \)? Wait, actually, the fourth side should complete the quadrilateral. Let's check the perimeter. Wait, no—Alex has three sides plotted, so the fourth side (from H to E or H to G) should make the total fencing (perimeter of the quadrilateral) ≤ 16. Wait, no, the three sides are E-F, F-G, and we need to find H such that the fourth side (E-H or G-H) plus the existing three sides sum to ≤16? Wait, no, maybe it's a quadrilateral with three sides given, so the fourth side is from H to E (if H is connected to E) or H to G (if connected to G). Wait, let's calculate the length of the three existing sides: E-F is 2, F-G is 5, and let's assume the fourth side is E-H or G-H. Wait, no, maybe it's a parallelogram? Wait, no, let's check the options. Let's compute the distance from each option to E(1,5) and G(6,1), and see which one makes the total fencing (sum of all four sides) ≤16.

First, compute the length of E-F (2), F-G (5), and then for each option H:

Option 1: (0,1)

• Distance from H(0,1) to E(1,5): \( \sqrt{(1 - 0)^2 + (5 - 1)^2} = \sqrt{1 + 16} = \sqrt{17} ≈ 4.123 \)
• Distance from H(0,1) to G(6,1): \( 6 - 0 = 6 \) (horizontal line, y=1)
• Total fencing: 2 (E-F) + 5 (F-G) + 6 (G-H) + 4.123 (H-E) ≈ 17.123 > 16. Not good.

Option 2: (0,-2)

• Distance from H(0,-2) to E(1,5): \( \sqrt{1 + 49} = \sqrt{50} ≈ 7.071 \)
• Distance from H(0,-2) to G(6,1): \( \sqrt{36 + 9} = \sqrt{45} ≈ 6.708 \)
• Total: 2 + 5 + 7.071 + 6.708 ≈ 20.779 > 16. No.

Option 3: (1,1)

• Distance from H(1,1) to E(1,5): \( 5 - 1 = 4 \) (vertical line, x=1)
• Distance from H(1,1) to G(6,1): \( 6 - 1 = 5 \) (horizontal line, y=1)
• Total fencing: 2 (E-F) + 5 (F-G) + 5 (G-H) + 4 (H-E) = 2 + 5 + 5 + 4 = 16. Perfect, that's exactly 16.

Option 4: (1,-2)

• Distance from H(1,-2) to E(1,5): \( 5 - (-2) = 7 \)
• Distance from H(1,-2) to G(6,1): \( \sqrt{25 + 9} = \sqrt{34} ≈ 5.830 \)
• Total: 2 + 5 + 7 + 5.830 ≈ 19.830 > 16. No.

So the correct answer is (1,1), which is option (1,1) (the third option in the list, assuming the options are (0,1), (0,-2), (1,1), (1,-2)). So the answer is (1,1).

Wait, but the options are:

• (0,1)
• (0,-2)
• (1,1)
• (1,-2)

So the correct one is (1,1).