Interpret geometric figures on grids - TACHS Math

1. The figure below shows polygon $ABCDEF$ on a grid. What is the area of this polygon, in square units?
   Answer: $22$ The L-shaped hexagon can be decomposed into two rectangles: a bottom rectangle with dimensions $6 \times 3 = 18$ square units, and a top-left rectangle with dimensions $4 \times 1 = 4$ square units. Total area $= 18 + 4 = 22$ square units.
2. Look at the figure shown on the grid. Polygon $WXYZ$ is reflected over the line $y=x$ to form polygon $W'X'Y'Z'$. What are the coordinates of $Y'$?
   Answer: $(4, -1)$ Reflection over $y=x$ swaps coordinates: $(a,b) \to (b,a)$. $Y(-1,4) \to Y'(4,-1)$. (A) is reflection over y-axis. (B) is reflection over $y=-x$. (C) is the original point's negation only in x.
3. Refer to the figure on the grid. Point $P$ lies on segment $AB$ such that $AP:PB = 3:1$. What are the coordinates of point $P$?
   Answer: $(5, 4)$ With $A(-1,-2)$ and $B(7,6)$, $P$ divides $AB$ in ratio $3:1$ from $A$. Using the section formula: $P=\left(\frac{3(7)+1(-1)}{4},\frac{3(6)+1(-2)}{4}\right)=\left(\frac{20}{4},\frac{16}{4}\right)=(5,4)$. (A) gives the midpoint. (C) uses ratio $1:1$ wrongly. (D) uses ratio $1:3$ from A.
4. Refer to the figure. What is the perimeter of the shaded L-shaped region?
   Answer: 14 units Tracing the outline from the lower-left corner: 4+1+2+2+2+3 = 14 units. Choices 12 and 13 miss one or two segments; 16 double-counts two edges.
5. Refer to the grid shown. Triangle $ABC$ has vertices at $A(-3, 4)$, $B(5, 4)$, and $C(2, -2)$. What is the area of triangle $ABC$ in square units?
   Answer: $24$ The base $AB$ is horizontal from $x=-3$ to $x=5$, so its length is $8$. The height is the perpendicular distance from $C(2,-2)$ to the line $y=4$, which is $|4-(-2)|=6$. Area $= \frac{1}{2}(8)(6) = 24$. (B) incorrectly uses base $10$. (C) forgets to multiply by $\frac{1}{2}$. (D) uses height $5$ from miscounting.
6. In the figure below, triangle $RST$ has been dilated from the origin by a scale factor of $\frac{3}{2}$ to create triangle $R'S'T'$. What are the coordinates of $T'$?
   Answer: $(6, -9)$ A dilation from the origin by factor $k$ maps $(x,y)\to(kx,ky)$. $T(4,-6)\to(4\cdot\frac{3}{2},-6\cdot\frac{3}{2})=(6,-9)$. (A) only scaled x by $\frac{3}{4}$. (C) scaled incorrectly. (D) scaled only x-coordinate.
7. Refer to the coordinate grid shown. Triangle $ABC$ is translated by the rule $(x,y)\to(x-3,y+2)$, and then reflected over the x-axis. What are the coordinates of the final image of vertex $C$?
   Answer: $(0, -5)$ Start with $C(3,3)$. Translation: $(3-3,3+2)=(0,5)$. Reflection over x-axis: $(x,y)\to(x,-y)$ gives $(0,-5)$. (A) forgets the reflection. (C) reflects over y-axis instead. (D) reverses translation direction.
8. Based on the figure, trapezoid $ABCD$ has $AB \parallel CD$. What is the area of trapezoid $ABCD$?
   Answer: $21$ square units From the figure, the parallel bases are $AB = 3$ units and $DC = 4$ units. The height (perpendicular distance between the parallel bases) is $3 - (-3) = 6$ units. Using the trapezoid area formula: Area $= \frac{1}{2}(b_1 + b_2) \times h = \frac{1}{2}(3 + 4) \times 6 = \frac{1}{2}(7)(6) = 21$ square units.

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