Draw Polygons on Coordinate Plane - 6th Grade Math

Points $V(3, -2)$ , $W(3, 4)$ , $X(-1, 4)$ , and $Y(-1, -2)$ form quadrilateral $VWXY$ . If point $Z$ is placed at the intersection of the quadrilateral's diagonals, what are the coordinates of point $Z$ ?
Answer: $(1, 1)$ , found by averaging the coordinates of opposite vertices of the rectangle When you encounter a problem about finding where diagonals intersect in a quadrilateral, start by identifying what type of quadrilateral you're working with. Looking at the coordinates $V(3, -2)$ , $W(3, 4)$ , $X(-1, 4)$ , and $Y(-1, -2)$ , notice that points $V$ and $W$ have the same $x$ -coordinate (3), and points $X$ and $Y$ have the same $x$ -coordinate (-1). Similarly, $W$ and $X$ share $y$ -coordinate (4), while $V$ and $Y$ share $y$ -coordinate (-2). This means you have a rectangle with sides parallel to the coordinate axes. In any rectangle, the diagonals intersect at the center point, which you can find by averaging the coordinates of opposite vertices. Taking opposite vertices $V(3, -2)$ and $X(-1, 4)$ : $x$ -coordinate = $\frac{3 + (-1)}{2} = 1$ , and $y$ -coordinate = $\frac{-2 + 4}{2} = 1$ . This gives you point $(1, 1)$ . Choice A incorrectly calls $(2, 1)$ the midpoint of diagonal $VX$ , but the actual midpoint calculation shows this is wrong. Choice C gives $(1, 2)$ , which reverses the coordinates. Choice D claims diagonals always intersect at the origin, which is completely false—the intersection point depends on the quadrilateral's position. Study tip: For rectangle problems, remember that the diagonals always intersect at the center, which you can find by averaging the coordinates of any pair of opposite vertices. This shortcut saves time compared to finding equations of diagonal lines.
A quadrilateral is drawn with vertices at $W(-2, -1)$ , $X(3, -1)$ , $Y(3, 3)$ , and $Z(-2, 3)$ . If point $P$ is placed at $(0, 1)$ , what is the relationship between point $P$ and quadrilateral $WXYZ$ ?
Answer: Point $P$ is inside the quadrilateral but not at its geometric center First, we identify that $WXYZ$ forms a rectangle with vertices at $W(-2, -1)$ , $X(3, -1)$ , $Y(3, 3)$ , and $Z(-2, 3)$ . The rectangle extends from $x = -2$ to $x = 3$ and from $y = -1$ to $y = 3$ . Point $P(0, 1)$ has coordinates within these ranges, so it's inside the rectangle. The center of the rectangle is at $\left(\frac{-2+3}{2}, \frac{-1+3}{2}\right) = (0.5, 1)$ . Since $P$ is at $(0, 1)$ , it's inside but not at the center. Choice A is wrong because $P$ is inside. Choice B is wrong because $P$ is not at the center $(0.5, 1)$ . Choice D is wrong because $P$ is not on any side of the rectangle.
A parallelogram has vertices $A(0,0)$ , $B(6,0)$ , $C(8,3)$ , and $D(2,3)$ , connected in order and closed. What is the length of the top side $\overline{DC}$ ?
Answer: 6 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂-x₁| if y-coordinates are the same, |y₂-y₁| if x-coordinates are the same). To draw the parallelogram, plot the vertices as ordered pairs A(0,0), B(6,0), C(8,3), and D(2,3), connect consecutive vertices with line segments, and close the polygon by connecting the last back to the first. Finding side lengths for axis-aligned sides: the top side DC from D(2,3) to C(8,3) has the same y=3, so length = |8-2| = 6 using x-coordinate difference (note: direction from D to C or C to D doesn't change length); absolute value ensures positive length. For example, bottom side AB is also 6 units. The correct length of DC is 6 units. A common error is misidentifying which side is top or using diagonal distances instead. To calculate: confirm same y for horizontal, use |x₂-x₁|, and verify with plot.
A rectangle is formed by connecting the points $A(4,-3)$ , $B(4,2)$ , $C(-2,2)$ , and $D(-2,-3)$ in order. What is the length of side $\overline{BC}$ ?
Answer: 6 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂ - x₁| if y-coordinates are the same, |y₂ - y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs A(4,-3), B(4,2), C(-2,2), and D(-2,-3), connect consecutive vertices with line segments in order, and close the polygon. For side lengths of axis-aligned sides, horizontal side BC from B(4,2) to C(-2,2) has same y=2, length | -2 - 4| = 6 using x-difference; absolute value ensures positive length. For this rectangle, the length of side BC is 6 units, matching choice B. A common error is using y-difference for horizontal, getting |2-2|=0, or arithmetic like | -2 -4|=2. To draw accurately: (1) identify axes and scale, (2) plot each vertex, (3) connect in order, (4) verify closed. For side lengths: (1) identify horizontal, (2) use x-difference with absolute value, (3) calculate correctly, and apply to perimeter or area (here vertical |2 - (-3)|=5, area 6×5=30), avoiding plotting errors or not closing the polygon.
A right triangle has vertices $A(2,3)$ , $B(6,3)$ , and $C(2,7)$ . What is the length of the horizontal leg $\overline{AB}$ ?
Answer: 4 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths with formulas like |x₂ - x₁| if y-coordinates are the same or |y₂ - y₁| if x-coordinates are the same. To draw the right triangle, plot the vertices as ordered pairs A(2,3), B(6,3), and C(2,7), connect consecutive vertices with line segments in order, and close the polygon by connecting the last point back to the first. For side lengths, the horizontal leg AB from (2,3) to (6,3) has the same y=3, so length = |6-2| = 4 units using the x-difference, while vertical sides like AC from (2,3) to (2,7) have the same x=2, so length = |7-3| = 4 units, with absolute values ensuring positive lengths. The length of the horizontal leg AB is 4 units. A common error is using y-differences for horizontal sides, such as |3-3| = 0, or arithmetic mistakes like 6-2=3. To draw accurately: (1) set up axes, (2) plot points, (3) connect and close, (4) confirm right angle. For lengths: (1) identify horizontal, (2) calculate |x₂ - x₁|, noting applications like area (1/2 × base × height = 1/2 × 4 × 4 = 8), while hypotenuse uses advanced methods, and avoiding errors like coordinate mix-ups.
A rectangle has vertices $W(-2,-2)$ , $X(3,-2)$ , $Y(3,4)$ , and $Z(-2,4)$ . Which pair of points forms a horizontal side of the rectangle?
Answer: $W(-2,-2)$ and $X(3,-2)$ This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂ - x₁| if y-coordinates are the same, |y₂ - y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs W(-2,-2), X(3,-2), Y(3,4), and Z(-2,4), connect consecutive vertices with line segments, and close the polygon. For identifying sides, horizontal sides have the same y-coordinate, like W(-2,-2) and X(3,-2) both at y=-2, length |3 - (-2)|=5 using x-difference; vertical sides have same x, like X to Y at x=3. The pair that forms a horizontal side is W(-2,-2) and X(3,-2), matching choice A. A common error is picking a diagonal pair like W and Y, which have different y and x, or misplotting coordinates reversed. To draw accurately: (1) identify axes and scale, (2) plot vertices, (3) connect in order, (4) verify closed. For side identification: (1) check coordinates for same y (horizontal) or x (vertical), (2) confirm with differences, and apply to perimeter (here 5+6+5+6=22 units) or area (5×6=30), avoiding mistakes like not recognizing axis-aligned sides.
A parallelogram has vertices $A(0,0)$ , $B(4,0)$ , $C(5,3)$ , and $D(1,3)$ . Which statement is true about the lengths of the horizontal sides $\overline{AB}$ and $\overline{CD}$ ?
Answer: $\overline{AB}=4$ units and $\overline{CD}=4$ units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths with formulas like |x₂ - x₁| if y-coordinates are the same or |y₂ - y₁| if x-coordinates are the same. To draw the parallelogram, plot the vertices as ordered pairs A(0,0), B(4,0), C(5,3), and D(1,3), connect consecutive vertices with line segments in order, and close the polygon by connecting the last point back to the first. For side lengths, the horizontal sides AB from (0,0) to (4,0) have the same y=0, so length = |4-0| = 4 units, and CD from (5,3) to (1,3) has the same y=3, so length = |1-5| = 4 units, with absolute values ensuring positive lengths. Thus, AB = 4 units and CD = 4 units is the true statement. A common error is misordering points or calculating non-horizontal sides as horizontal, leading to incorrect lengths like 5 or 3 units. To draw accurately: (1) identify axes, (2) plot vertices, (3) connect in order, and (4) verify shape. For lengths: (1) check for same y for horizontal, (2) use |x₂ - x₁|, noting parallelogram properties, while diagonals need Pythagorean theorem, and avoiding mistakes like no absolute value or arithmetic errors.
A student wants to draw a rectangle on a coordinate plane using the points $A(-5,1)$ , $B(1,1)$ , $C(1,6)$ , and $D(-5,6)$ . Which pair of points forms a vertical side, and what is its length?
Answer: $\overline{BC}$ , length 5 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂-x₁| if y-coordinates are the same, |y₂-y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs A(-5,1), B(1,1), C(1,6), and D(-5,6), connect consecutive vertices with line segments, and close the polygon by connecting the last back to the first. Finding side lengths for axis-aligned sides: vertical sides BC from (1,1) to (1,6) and DA from (-5,6) to (-5,1) have same x, length = |6-1| = 5; horizontal sides have same y, length = |1 - (-5)| = 6; absolute value ensures positive length. For example, BC is vertical with length 5 units. The correct pair is BC with length 5 units, as it is vertical. A common error is confusing vertical and horizontal or miscalculating differences. To identify: check for same x (vertical), calculate |y₂-y₁|, and match to choices.
A small garden is mapped on a coordinate plane with corners at $(-1,2)$ , $(6,2)$ , $(6,5)$ , and $(-1,5)$ . If you connect the points in that order to make a rectangle, what is the length of the side from $(-1,2)$ to $(6,2)$ ?
Answer: 7 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂ - x₁| if y-coordinates are the same, |y₂ - y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs (-1,2), (6,2), (6,5), and (-1,5), connect consecutive vertices with line segments, and close the polygon by connecting the last back to the first. For side lengths of axis-aligned sides, the horizontal side from (-1,2) to (6,2) has the same y-coordinate of 2, so length |6 - (-1)| = 7 using the x-difference, while a vertical side like (6,2) to (6,5) has the same x-coordinate of 6, length |5 - 2| = 3 using the y-difference; absolute value ensures positive length. For this garden rectangle, the correct side length from (-1,2) to (6,2) is 7 units, matching choice B. A common error is using the y-difference for a horizontal side instead of x-difference, such as |2 - 2| = 0, or forgetting absolute value to get a negative length like 6 - (-1) = 7 but miscalculating arithmetic as 5. To draw accurately: (1) identify axes and scale, (2) plot each vertex by counting x units right or left and y units up or down, (3) connect in order, and (4) verify closed. For side lengths: (1) check if horizontal or vertical, (2) use the right difference with absolute value, (3) calculate correctly, and apply to perimeter (here, 7 + 3 + 7 + 3 = 20 units) or area (7 × 3 = 21 square units), avoiding mistakes like reversing coordinates or not closing the polygon.
A right triangle is formed with vertices at $R(0, 0)$ , $S(8, 0)$ , and $T(0, 6)$ . If a rectangle is drawn using this triangle's legs as two of its sides, and the rectangle shares vertices $R$ and $S$ with the triangle, what is the area of the rectangle?
Answer: $48$ square units, using the legs of the right triangle as adjacent sides of the rectangle When you encounter problems involving right triangles and rectangles on a coordinate plane, start by identifying the key measurements and visualizing how the shapes relate to each other. Looking at the given vertices, triangle $RST$ has its right angle at $R(0,0)$ , with one leg extending horizontally from $R$ to $S(8,0)$ and another leg extending vertically from $R$ to $T(0,6)$ . The horizontal leg has length $8$ units, and the vertical leg has length $6$ units. Since the rectangle shares vertices $R$ and $S$ with the triangle and uses the triangle's legs as two of its sides, the rectangle must have dimensions $8 \times 6$ . Therefore, its area is $8 \times 6 = 48$ square units. Choice A incorrectly suggests adding areas together, but the question asks for the rectangle's area alone, not a combined area. Choice B makes an error about the relationship between the rectangle and triangle areas - the rectangle actually has twice the triangle's area ( $48$ vs $24$ ), not half of some other rectangle's area. Choice C miscalculates by seemingly squaring one dimension ( $8^2 = 64$ ), but rectangles require multiplying length times width, not squaring a single measurement. The correct answer is D: $48$ square units, calculated by multiplying the lengths of the triangle's legs. Study tip: When working with coordinate geometry problems, always plot the points or sketch the figure first. This helps you visualize relationships between shapes and avoid calculation errors.
Triangle $ABC$ is plotted with vertices $A(-4, 2)$ , $B(2, 2)$ , and $C(-1, 6)$ . If the triangle is reflected across the line $x = -1$ to form triangle $A'B'C'$ , what is the length of side $A'B'$ in the reflected triangle?
Answer: $6$ units, since reflections preserve distances and side $AB$ has this length When reflecting across $x = -1$ , point $A(-4, 2)$ reflects to $A'(2, 2)$ because it's $3$ units left of the line, so it goes $3$ units right: $-1 + 3 = 2$ . Point $B(2, 2)$ reflects to $B'(-4, 2)$ because it's $3$ units right of the line, so it goes $3$ units left: $-1 - 3 = -4$ . Point $C(-1, 6)$ reflects to $C'(-1, 6)$ because it's on the line of reflection. Now $A'B'$ connects $(2, 2)$ and $(-4, 2)$ . Since both points have the same $y$ -coordinate, the distance is $|2 - (-4)| = 6$ units. This is the same as the original length $AB$ , confirming that reflections preserve distances. Choice B incorrectly suggests distances change. Choice C gives an incorrect calculation. Choice D incorrectly applies the distance formula.
Plot and connect the points $P(1,-4)$ , $Q(1,2)$ , $R(5,2)$ , and $S(5,-4)$ in the order $P\to Q\to R\to S\to P$ . What is the area of the rectangle in square units?
Answer: 24 square units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂ - x₁| if y-coordinates are the same, |y₂ - y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs P(1,-4), Q(1,2), R(5,2), and S(5,-4), connect consecutive vertices with line segments in the order P to Q to R to S back to P, and close the polygon. For side lengths of axis-aligned sides, vertical sides like P to Q have the same x-coordinate of 1, length |2 - (-4)| = 6 using the y-difference, and horizontal sides like Q to R have the same y-coordinate of 2, length |5 - 1| = 4 using the x-difference; absolute value ensures positive length. For this rectangle, plot and connect to find lengths 6 (vertical), 4 (horizontal), 6 (vertical), and 4 (horizontal), so the area is base × height = 4 × 6 = 24 square units, matching choice A. A common error is plotting coordinates reversed, like treating (1,-4) as (-4,1), or using x-difference for vertical sides, leading to wrong area like 6 × 6 = 36. To draw accurately: (1) identify axes and scale, (2) plot each vertex carefully, (3) connect in order, and (4) verify closed. For applications like area: calculate side lengths first, then multiply for rectangles, while for perimeter sum them (here 6 + 4 + 6 + 4 = 20 units), avoiding arithmetic errors or forgetting to close the polygon.
You plot points $A(-4,2)$ , $B(-4,-3)$ , $C(1,-3)$ , and $D(1,2)$ and connect them in order. What is the length of the vertical side from $A(-4,2)$ to $B(-4,-3)$ ?
Answer: 5 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂ - x₁| if y-coordinates are the same, |y₂ - y₁| if x-coordinates are the same). To draw the polygon, plot the vertices as ordered pairs A(-4,2), B(-4,-3), C(1,-3), and D(1,2), connect consecutive vertices with line segments, and close the polygon. For side lengths of axis-aligned sides, the vertical side from A to B has the same x-coordinate of -4, length | -3 - 2| = 5 using the y-difference; absolute value ensures positive length. For this shape, the length of the vertical side from A(-4,2) to B(-4,-3) is 5 units, matching choice C. A common error is forgetting absolute value, getting -5, or using x-difference for vertical, like | -4 - (-4)|=0. To draw accurately: (1) identify axes and scale, (2) plot each vertex, (3) connect in order, (4) verify closed. For side lengths: (1) identify vertical, (2) use y-difference with absolute value, (3) calculate correctly, and apply to perimeter or area, avoiding arithmetic errors or plotting mistakes.
A skateboard ramp is outlined on a coordinate plane by the points $A(-3,2)$ , $B(4,2)$ , $C(4,-1)$ , and $D(-3,-1)$ . The ramp is a rectangle. What is the length of side $\overline{AB}$ ?
Answer: 7 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths with formulas like |x₂ - x₁| if y-coordinates are the same or |y₂ - y₁| if x-coordinates are the same. To draw the rectangle, plot the vertices as ordered pairs A(-3,2), B(4,2), C(4,-1), and D(-3,-1), connect consecutive vertices with line segments in order, and close the polygon by connecting the last point back to the first. For side lengths, the horizontal side AB from (-3,2) to (4,2) has the same y=2, so length = |4 - (-3)| = 7 units using the x-difference, while vertical sides like BC from (4,2) to (4,-1) have the same x=4, so length = |-1 - 2| = 3 units using the y-difference, with absolute values ensuring positive lengths. For this ramp rectangle, the correct length of side AB is 7 units. A common error is forgetting to add the absolute value for negative coordinates, such as calculating 4 - (-3) as 1 instead of 7, or using y-differences for horizontal sides. To draw accurately: (1) identify the axes and scale, including negative values, (2) plot each vertex carefully, (3) connect in order, and (4) verify closure. For side lengths: (1) identify horizontal or vertical alignment, (2) use the appropriate difference, (3) calculate with absolute value, noting applications like material needs for the ramp, and avoiding mistakes like coordinate reversal or arithmetic errors.
A right triangle is made by plotting $A(2,3)$ , $B(8,3)$ , and $C(2,7)$ and connecting the points. What is the length of the horizontal side $\overline{AB}$ ?
Answer: 6 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂-x₁| if y-coordinates are the same, |y₂-y₁| if x-coordinates are the same). To draw the right triangle, plot the vertices as ordered pairs A(2,3), B(8,3), and C(2,7), connect consecutive vertices with line segments, and close the polygon by connecting the last back to the first. Finding side lengths for axis-aligned sides: the horizontal side AB from (2,3) to (8,3) has the same y=3, so length = |8-2| = 6 using x-coordinate difference; absolute value ensures positive length. For example, vertical side AC would be |7-3| = 4 units. The correct length of AB is 6 units. A common error is using y-differences for horizontal sides or forgetting absolute value. To calculate: identify horizontal (same y), use |x₂-x₁|, and compute.
A rectangle has vertices $A(1,-3)$ , $B(6,-3)$ , $C(6,2)$ , and $D(1,2)$ . What is the area of the rectangle in square units?
Answer: 25 square units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths with formulas like |x₂ - x₁| if y-coordinates are the same or |y₂ - y₁| if x-coordinates are the same, and calculating area. To draw the rectangle, plot the vertices as ordered pairs A(1,-3), B(6,-3), C(6,2), and D(1,2), connect consecutive vertices with line segments in order, and close the polygon by connecting the last point back to the first. For side lengths, horizontal sides like AB from (1,-3) to (6,-3) have the same y=-3, so length = |6-1| = 5 units, while vertical sides like BC from (6,-3) to (6,2) have the same x=6, so length = |2 - (-3)| = 5 units. The area of the rectangle is 5 × 5 = 25 square units. A common error is miscalculating height as 2 - 3 = -1 or forgetting the negative, leading to areas like 20 or 15. To draw accurately: (1) include negative y on axes, (2) plot vertices, (3) connect, (4) verify. For area: (1) find lengths, (2) multiply, noting uses like coverage, while diagonals are advanced, and avoiding mistakes like no absolute value or wrong differences.
A bulletin board is drawn on a coordinate plane as a rectangle with corners at $(2,1)$ , $(7,1)$ , $(7,5)$ , and $(2,5)$ . What is the perimeter of the rectangle in units?
Answer: 18 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂ - x₁| if y-coordinates are the same, |y₂ - y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs (2,1), (7,1), (7,5), and (2,5), connect consecutive vertices with line segments, and close the polygon. For side lengths of axis-aligned sides, horizontal sides have the same y, length |7 - 2| = 5 using x-difference, and vertical sides have the same x, length |5 - 1| = 4 using y-difference; absolute value ensures positive length. For this bulletin board rectangle, lengths are 5, 4, 5, 4, so perimeter is 5 + 4 + 5 + 4 = 18 units, matching choice A. A common error is arithmetic like |7-2|=4, or using y for horizontal leading to wrong perimeter. To draw accurately: (1) identify axes and scale, (2) plot vertices, (3) connect in order, (4) verify closed. For perimeter: sum calculated side lengths, or for area 5 × 4 = 20 square units, avoiding mistakes like forgetting absolute value or not closing the polygon.
A rectangle is formed by connecting the points $(-3,4)$ , $(1,4)$ , $(1,-2)$ , and $(-3,-2)$ in that order. What is the length of the vertical side from $(1,4)$ to $(1,-2)$ ?
Answer: 6 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths (|x₂-x₁| if y-coordinates are the same, |y₂-y₁| if x-coordinates are the same). To draw the rectangle, plot the vertices as ordered pairs (-3,4), (1,4), (1,-2), and (-3,-2), connect consecutive vertices with line segments, and close the polygon by connecting the last back to the first. Finding side lengths for axis-aligned sides: the vertical side from (1,4) to (1,-2) has the same x=1, so length = |-2 - 4| = |-6| = 6 using y-coordinate difference; absolute value ensures positive length. For example, if points were (5,1) to (5,4), length = |4-1| = 3. The correct length is 6 units. A common error is not using absolute value, resulting in negative lengths like -6. To calculate vertical sides: check same x, use |y₂-y₁|, and compute positively.
A rectangle is formed by connecting the points $A(-2,4)$ , $B(3,4)$ , $C(3,-1)$ , and $D(-2,-1)$ in order. What is the length of the vertical side $\overline{BC}$ ?
Answer: 5 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths with formulas like |x₂ - x₁| if y-coordinates are the same or |y₂ - y₁| if x-coordinates are the same. To draw the rectangle, plot the vertices as ordered pairs A(-2,4), B(3,4), C(3,-1), and D(-2,-1), connect consecutive vertices with line segments in order, and close the polygon by connecting the last point back to the first. For side lengths, the vertical side BC from (3,4) to (3,-1) has the same x=3, so length = |-1 - 4| = 5 units using the y-difference, with absolute value ensuring positive length. The length of the vertical side BC is 5 units. A common error is calculating without absolute value or mixing with x-differences, leading to lengths like 4 or -5. To draw accurately: (1) handle negatives on axes, (2) plot points, (3) connect, (4) check closure. For lengths: (1) identify vertical, (2) use |y₂ - y₁|, noting rectangle properties like area (width |3 - (-2)|=5 × 5=25), and avoiding arithmetic or coordinate errors.
A rectangular garden has corners at $(2,1)$ , $(7,1)$ , $(7,5)$ , and $(2,5)$ on a coordinate plane. What is the perimeter of the garden in units?
Answer: 18 units This question tests drawing polygons on a coordinate plane given vertex coordinates, using coordinates to find horizontal and vertical side lengths with formulas like |x₂ - x₁| if y-coordinates are the same or |y₂ - y₁| if x-coordinates are the same. To draw the rectangular garden, plot the vertices as ordered pairs (2,1), (7,1), (7,5), and (2,5), connect consecutive vertices with line segments in order, and close the polygon by connecting the last point back to the first. For side lengths, horizontal sides like from (2,1) to (7,1) have the same y=1, so length = |7-2| = 5 units, while vertical sides like from (7,1) to (7,5) have the same x=7, so length = |5-1| = 4 units, with absolute values ensuring positive lengths. The perimeter of the garden is 2 × (5 + 4) = 18 units. A common error is calculating width as 6 or height as 5, leading to perimeters like 20 or 22 units. To draw accurately: (1) use grid scale, (2) plot points, (3) connect, (4) verify rectangle. For perimeter: (1) determine side types, (2) compute differences, (3) sum, noting uses like fencing, area 5×4=20, and avoiding mistakes like no absolute value or wrong subtraction.

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