Free Math Worksheets

- In an axis-aligned rectangle, which coordinates are shared by vertically or horizontally aligned vertices? - The midpoint coordinates are the averages of the corresponding endpoint coordinates. - Check the x-coordinates and y-coordinates separately.

1. In rectangle \(ABCD\), point \(D\) must have the same x-coordinate as \(A\) and the same y-coordinate as \(C\). Therefore, \(D = (2, 8)\). 2. Use the midpoint formula on \(A(2, 2)\) and \(B(10, 2)\): \(M = \left(\frac{2 + 10}{2}, \frac{2 + 2}{2}\right) = (6, 2)\).

a) \(D = (2, 8)\) b) \(M = (6, 2)\)

- Average the two x-coordinates to find the value halfway between them. - Compare both coordinates of \(S\) with the coordinates of \(Q\). - Change only the coordinate described in each condition.

1. The x-coordinate halfway between \(3\) and \(11\) is \(\frac{3 + 11}{2} = 7\). The y-coordinate is \(10\), so \(S = (7, 10)\). These are also the coordinates of \(Q\), so \(S\) and \(Q\) name the same point. 2. Point \(T\) has x-coordinate \(3\) and y-coordinate \(10 - 3 = 7\), so \(T = (3, 7)\).

a) \(S = (7, 10)\). It is the same point as \(Q\). b) \(T = (3, 7)\)

- For a midpoint, average both pairs of coordinates. - Translate “twice” into multiplication. - Determine the x-coordinate and y-coordinate separately.

1. Apply the midpoint formula to \(K(1, 1)\) and \(M(9, 9)\): \(Z = \left(\frac{1 + 9}{2}, \frac{1 + 9}{2}\right) = (5, 5)\). 2. The x-coordinate of \(W\) is \(\frac{1 + 9}{2} = 5\). Its y-coordinate is \(2 \cdot 9 = 18\). Therefore, \(W = (5, 18)\).

a) \(Z = (5, 5)\) b) \(W = (5, 18)\)

- Check whether each pair of points is aligned horizontally or vertically. - For aligned points, subtract the coordinates that differ. - The distance from a point to a line is measured along a perpendicular segment.

1. Points \(A\) and \(B\) have the same x-coordinate, so \(AB=|9-2|=7\) units. 2. Points \(A\) and \(C\) have the same y-coordinate, so \(AC=|6-1|=5\) units. 3. Line \(g\) is the vertical line \(x=1\). The perpendicular distance from \(C(6,2)\) to \(g\) is the horizontal difference \(|6-1|=5\) units.

a) \(7\) units b) \(5\) units c) \(5\) units

- Determine the equation of the horizontal line. - Points on the same vertical grid line have the same x-coordinate. - A point can be above or below a horizontal line and have the same distance from it.

1. Line \(h\) is horizontal with equation \(y=5\). 2. The distance from \(R(6,8)\) to \(h\) is \(|8-5|=3\) units. 3. Because \(S\) lies on the same vertical line as \(R\), its x-coordinate is \(6\). 4. A point \(2\) units from \(y=5\) can have y-coordinate \(5+2=7\) or \(5-2=3\). Thus, the two points are \(S_1(6,7)\) and \(S_2(6,3)\).

a) \(3\) units b) \(S_1(6, 7)\) and \(S_2(6, 3)\)

- Use the coordinates of two adjacent vertices to find a side length. - Write the horizontal or vertical line containing each side. - Measure each perpendicular distance by subtracting coordinates.

1. Side \(\overline{AB}\) is horizontal, so its length is \(|8-2|=6\) units. 2. The sides lie on \(y=2\), \(x=8\), \(y=8\), and \(x=2\). 3. From \(M(5,5)\), the perpendicular distances are \(|5-2|=3\), \(|8-5|=3\), \(|8-5|=3\), and \(|5-2|=3\) units. 4. Therefore, \(M\) is \(3\) units from every side.

a) \(6\) units b) \(3\) units from each side

- Determine the equation of line \(g\). - A perpendicular to a horizontal line is vertical. - The closest point on \(g\) has the same x-coordinate as \(S\).

1. Line \(g\) is horizontal with equation \(y=1\). The distance from \(S(6,5)\) to \(g\) is \(5-1=4\) units. 2. A perpendicular to a horizontal line is vertical, so \(F\) has the same x-coordinate as \(S\). Since \(F\) lies on \(g\), \(F=(6,1)\). 3. Points \(A(2,1)\) and \(F(6,1)\) lie on a horizontal line, so \(AF=6-2=4\) units.

a) \(4\) units b) \(F(6, 1)\) c) \(4\) units

- Match x- and y-coordinates to complete the axis-aligned rectangle. - Find the two side lengths before using the perimeter formula. - The diagonals of a rectangle meet at their common midpoint.

1. Point \(S\) has the same x-coordinate as \(P\) and the same y-coordinate as \(R\), so \(S = (1, 5)\). 2. The side lengths are \(9 - 1 = 8\,\text{cm}\) and \(5 - 1 = 4\,\text{cm}\). The perimeter is \(2(8 + 4) = 24\,\text{cm}\). 3. The diagonals of a rectangle bisect each other. The midpoint of \(P(1, 1)\) and \(R(9, 5)\) is \(M = \left(\frac{1 + 9}{2}, \frac{1 + 5}{2}\right) = (5, 3)\).

1. \(S = (1, 5)\) 2. \(24\,\text{cm}\) 3. \(M = (5, 3)\)

- Opposite sides of a parallelogram have equal direction vectors. - The midpoint coordinates are the averages of the endpoint coordinates. - Use the result from part b) when finding \(Q\).

1. The vector from \(B\) to \(C\) is \((2, 4)\). Apply the same vector to \(A\): \(D = (2 + 2, 1 + 4) = (4, 5)\). 2. The midpoint of \(A(2, 1)\) and \(B(8, 1)\) is \(P = \left(\frac{2 + 8}{2}, \frac{1 + 1}{2}\right) = (5, 1)\). 3. The midpoint of \(P(5, 1)\) and \(C(10, 5)\) is \(Q = \left(\frac{5 + 10}{2}, \frac{1 + 5}{2}\right) = (7.5, 3)\).

a) \(D = (4, 5)\) b) \(P = (5, 1)\) c) \(Q = (7.5, 3)\)

- First determine the missing vertex \(B\). - A horizontal segment’s midpoint has the same y-coordinate as its endpoints. - Average the x-coordinates of \(B\) and \(C\).

1. The fourth vertex of the rectangle is \(B = (8, 6)\). 2. Apply the midpoint formula to \(B(8, 6)\) and \(C(0, 6)\): \(M = \left(\frac{8 + 0}{2}, \frac{6 + 6}{2}\right) = (4, 6)\).

\(M = (4, 6)\)

- The diagonals of a rhombus bisect each other. - Place half of each diagonal length on both sides of the origin. - Follow the requested clockwise labeling order.

1. The diagonals of a rhombus bisect each other, so the origin is the midpoint of both diagonals. 2. Half of the horizontal diagonal is \(6\), giving endpoints \((-6, 0)\) and \((6, 0)\). 3. Half of the vertical diagonal is \(5\), giving endpoints \((0, 5)\) and \((0, -5)\). 4. Following the requested clockwise order gives \(A(-6, 0)\), \(B(0, 5)\), \(C(6, 0)\), and \(D(0, -5)\).

\(A(-6, 0)\), \(B(0, 5)\), \(C(6, 0)\), and \(D(0, -5)\)

- Use the midpoint formula for each segment. - Work one segment at a time. - Compare the two final coordinate pairs.

1. \(P = \left(\frac{0 + 2}{2}, \frac{0 + 6}{2}\right) = (1, 3)\), and \(Q = \left(\frac{8 + 10}{2}, \frac{6 + 0}{2}\right) = (9, 3)\). 2. \(M_1 = \left(\frac{0 + 10}{2}, \frac{0 + 0}{2}\right) = (5, 0)\), and \(M_2 = \left(\frac{2 + 8}{2}, \frac{6 + 6}{2}\right) = (5, 6)\). 3. The midpoint of \(\overline{PQ}\) is \(\left(\frac{1 + 9}{2}, \frac{3 + 3}{2}\right) = (5, 3)\). 4. The midpoint of \(\overline{M_1M_2}\) is \(\left(\frac{5 + 5}{2}, \frac{0 + 6}{2}\right) = (5, 3)\). 5. The two midpoints coincide.

1. \(P = (1, 3)\), \(Q = (9, 3)\) 2. \(M_1 = (5, 0)\), \(M_2 = (5, 6)\) 3. Both final midpoints are \((5, 3)\).