First, find the length and width of the rectangle. The length is the difference between the x-coordinates: 9 - 2 = 7 units. The width is the difference between the y-coordinates: 7 - 2 = 5 units. The area of a rectangle is length times width. So, the area is 7 * 5 = 35 square units.

Let's analyze the sides. The side from (2, 1) to (6, 1) is horizontal with length 4. The side from (4, 4) to (8, 4) is also horizontal with length 4. These sides are parallel. The side from (6, 1) to (8, 4) moves 2 units right and 3 units up. The side from (2, 1) to (4, 4) also moves 2 units right and 3 units up. Since both pairs of opposite sides are parallel and equal in length, the shape is a parallelogram. It is not a rectangle because the adjacent sides are not perpendicular.

A right triangle with two equal sides is an isosceles right triangle. We need to find the lengths of the two sides that form the right angle. For choice A, the points are (1, 2), (1, 6), and (5, 2). The vertical side from (1, 2) to (1, 6) has length 6 - 2 = 4. The horizontal side from (1, 2) to (5, 2) has length 5 - 1 = 4. Since these two sides are equal and form a right angle, this is an isosceles right triangle.

The playground's x-values range from 2 to 12, and its y-values range from 3 to 10. A point is 'inside' if its x-coordinate is strictly between 2 and 12, and its y-coordinate is strictly between 3 and 10. The point (8, 8) satisfies these conditions because 2 < 8 < 12 and 3 < 8 < 10. The other points are all on the border: (12, 5) is on the right edge, (7, 10) is on the top edge, and (2, 3) is a corner.

The given square has a side length of 5 - 1 = 4 units. When an identical square is placed side-by-side, they share one side. Let's place the second square to the right of the first. Its vertices would be (5, 1), (9, 1), (9, 5), and (5, 5). The new, larger rectangle's vertices are (1, 1), (9, 1), (9, 5), and (1, 5). The length of this rectangle is 9 - 1 = 8 units, and the width is 5 - 1 = 4 units. The perimeter is 2 * (length + width) = 2 * (8 + 4) = 2 * 12 = 24 units.

When you see a coordinate grid problem involving distance, you're dealing with movement along straight lines between points. The key is to calculate the distance for each segment of the journey separately, then add them together.

Let's trace the ant's path step by step. The ant starts at (3, 1) and walks to (3, 5). Notice that the x-coordinate stays the same (3), so this is a vertical movement. The distance is the difference in y-coordinates: $$5 - 1 = 4$$ units.

Next, the ant walks from (3, 5) to (9, 5). Now the y-coordinate stays the same (5), so this is a horizontal movement. The distance is the difference in x-coordinates: $$9 - 3 = 6$$ units.

The total distance is $$4 + 6 = 10$$ units, which is answer D.

Let's examine why the other choices are wrong. Choice A (2 units) is far too small—it might represent a single coordinate difference rather than the full journey. Choice B (24 units) could result from multiplying distances instead of adding them ($$4 \times 6 = 24$$). Choice C (12 units) might come from incorrectly calculating one of the segments or adding an extra step.

Remember this strategy: for grid problems involving right-angle movements, calculate each segment separately by finding the difference in coordinates that change, then add all segments together. Always double-check that you're subtracting coordinates correctly (larger minus smaller) to get positive distances.

When you see a question about finding the midpoint of a line segment, you're working with coordinate geometry. The midpoint is simply the point that sits exactly halfway between two endpoints.

To find the midpoint, you use the midpoint formula: take the average of the x-coordinates and the average of the y-coordinates. For points $$(x_1, y_1)$$ and $$(x_2, y_2)$$, the midpoint is $$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$.

With endpoints (2, 7) and (10, 7), let's calculate:

• x-coordinate of midpoint: $$\frac{2 + 10}{2} = \frac{12}{2} = 6$$
• y-coordinate of midpoint: $$\frac{7 + 7}{2} = \frac{14}{2} = 7$$

So the midpoint is (6, 7), which is answer choice D.

Now let's see why the other answers are wrong. Choice A gives (5, 7) - this incorrectly calculates the x-coordinate as 5 instead of 6, perhaps from adding 2 + 10 = 12 but then dividing by something other than 2. Choice B gives (6, 14) - this correctly finds the x-coordinate as 6 but mistakes the y-coordinate as 14, which would happen if you added the y-coordinates (7 + 7 = 14) but forgot to divide by 2. Choice C gives (8, 7) - this gets the y-coordinate right but miscalculates the x-coordinate, possibly from finding the difference (10 - 2 = 8) rather than the average.

Remember: for midpoints, always average both coordinates separately. When both endpoints have the same y-coordinate (like here), you're dealing with a horizontal line, so the midpoint will have that same y-coordinate.

When you encounter a question about distance from a point to the sides of a square, you need to visualize the square and determine which side is closest, then calculate the perpendicular distance.

First, let's plot the square with corners at (4, 4), (4, 9), (9, 9), and (9, 4). This creates a square with sides along the lines: bottom side from (4, 4) to (9, 4), top side from (4, 9) to (9, 9), left side from (4, 4) to (4, 9), and right side from (9, 4) to (9, 9). The point (6, 6) lies inside this square.

Since (6, 6) is inside the square, we need the shortest perpendicular distance to any side. The bottom side lies along the line $$y = 4$$, so the distance is $$|6 - 4| = 2$$ units. The top side lies along $$y = 9$$, giving distance $$|9 - 6| = 3$$ units. The left side lies along $$x = 4$$, giving distance $$|6 - 4| = 2$$ units. The right side lies along $$x = 9$$, giving distance $$|9 - 6| = 3$$ units.

The closest sides are the bottom and left sides, both 2 units away, making D correct.

Choice A (1 unit) might tempt you if you miscalculate the coordinates. Choice B (5 units) is the side length of the square, not a distance to a side. Choice C (3 units) represents the distance to the top or right sides, which are farther away.

Remember: when finding distance from a point to the side of a rectangle or square, calculate the perpendicular distance to each side and choose the minimum.

The small square has a vertex at (0, 0) and is inside the larger rectangle. This means its sides must extend along the positive x-axis and positive y-axis. Since the side length is 3, the vertices of the square are (0, 0), (3, 0), (0, 3), and (3, 3). The vertex that is diagonally opposite from (0, 0) is the one that shares neither its x- nor y-coordinate, which is (3, 3).

The vertices of the shape must be (3, 5), (8, 5), (8, 10), and (3, 10). Let's find the side lengths. The horizontal distance between the x-coordinates is 8 - 3 = 5 units. The vertical distance between the y-coordinates is 10 - 5 = 5 units. Since the lengths of the adjacent sides are equal (both are 5 units) and the sides are horizontal and vertical, the shape is a square.