Theorems about Parallelograms - Geometry

1. Triangle $ABC$ has vertices $A$, $B$, and $C$. The centroid $G$ is located at coordinates $(4, 3)$. If vertex $A$ is at $(1, 6)$ and vertex $B$ is at $(5, 2)$, what are the coordinates of vertex $C$?
   Answer: $(6, 1)$ using the centroid formula for coordinate geometry The centroid is located at $\left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right)$. Given $G(4,3)$, $A(1,6)$, and $B(5,2)$: For x-coordinate: $\frac{1 + 5 + x_C}{3} = 4$, so $6 + x_C = 12$, thus $x_C = 6$. For y-coordinate: $\frac{6 + 2 + y_C}{3} = 3$, so $8 + y_C = 9$, thus $y_C = 1$. Therefore $C = (6,1)$. Choice B uses incorrect y-calculation. Choice C doubles the x-coordinate. Choice D uses the sum instead of solving for the unknown coordinate.
2. In triangle $PQR$, the three medians intersect at point $G$. If median $PS$ has length $15$, where $S$ is the midpoint of $QR$, what is the distance from $G$ to vertex $P$?
   Answer: $10$ units from $P$ along the median to the centroid The medians of a triangle meet at the centroid, which divides each median in a 2:1 ratio, with the longer segment being from the vertex to the centroid. Since median $PS = 15$, the distance from $P$ to centroid $G$ is $\frac{2}{3} \cdot 15 = 10$ units. Choice A gives $\frac{1}{3}$ of the median length. Choice B gives half the median length. Choice D gives $\frac{4}{5}$ of the median length, incorrectly applying a different ratio.
3. In triangle $ABC$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $AC$. If $BC = 18$ and $\angle BAC = 50°$, what can be concluded about the relationship between segment $MN$ and side $BC$?
   Answer: $MN$ is parallel to $BC$ and $MN = 9$, with $\angle AMN = 50°$ By the midpoint theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore, $MN \parallel BC$ and $MN = \frac{1}{2} \cdot 18 = 9$. Since $MN \parallel BC$, corresponding angles are equal, so $\angle AMN = \angle BAC = 50°$. Choice B incorrectly states $MN = 18$ (should be half). Choice C incorrectly claims perpendicularity and wrong angle. Choice D incorrectly calculates the angle as supplementary.
4. In triangle ABC, point D lies on side BC such that AD bisects ∠BAC. If AB = 12 and AC = 15, what is the ratio BD:DC?
   Answer: 4:5 by the angle bisector theorem and proportional segments By the angle bisector theorem, when AD bisects ∠BAC, the ratio BD:DC = AB:AC = 12:15 = 4:5. Choice B reverses the ratio. Choices C and D use incorrect proportions that don't match the given side lengths.
5. Triangle $JKL$ has medians $JM$, $KN$, and $LO$ that intersect at centroid $G$. If $JM = 21$ and $KN = 18$, and the distance from $K$ to $G$ is $12$, what is the distance from $G$ to the midpoint $M$ of side $KL$?
   Answer: $7$ units using centroid division properties of medians When you encounter problems involving medians and centroids, remember that the centroid divides each median in a specific 2:1 ratio, with the longer segment extending from the vertex to the centroid. The centroid $G$ divides every median so that the distance from any vertex to $G$ is twice the distance from $G$ to the opposite midpoint. Since the distance from $K$ to $G$ is 12, and $G$ divides median $KN$ in a 2:1 ratio, the distance from $G$ to midpoint $N$ must be $12 ÷ 2 = 6$. We can verify this using the given information: if $KG = 12$ and $GN = 6$, then the total length $KN = 12 + 6 = 18$, which matches the given median length. Now, for median $JM$ with length 21, the centroid divides it the same way. The distance from $J$ to $G$ is $\frac{2}{3} × 21 = 14$, and the distance from $G$ to midpoint $M$ is $\frac{1}{3} × 21 = 7$. Choice A gives 6 units, which incorrectly applies the distance from $G$ to $N$ instead of finding the distance to $M$. Choice B gives 14 units, which represents the distance from $J$ to $G$, not from $G$ to $M$. Choice C gives 9 units, which appears to incorrectly calculate half of the median length $JM$. Study tip: Always remember the centroid's 2:1 division rule: vertex to centroid is $\frac{2}{3}$ of the median length, while centroid to midpoint is $\frac{1}{3}$ of the median length.
6. Triangle $DEF$ is isosceles with $DE = DF$. If the measure of the exterior angle at vertex $E$ is $110°$, what is the measure of $\angle EDF$?
   Answer: $40°$ because each base angle measures $70°$ The exterior angle at $E$ is $110°$, so the interior angle at $E$ is $180° - 110° = 70°$. Since triangle $DEF$ is isosceles with $DE = DF$, the base angles $\angle DEF$ and $\angle DFE$ are congruent, each measuring $70°$. Using the triangle angle sum theorem: $\angle EDF + 70° + 70° = 180°$, so $\angle EDF = 40°$. Choice A incorrectly divides the base angle sum by 2. Choice C confuses the exterior angle with a base angle. Choice D incorrectly applies the exterior angle relationship.
7. In triangle $XYZ$, $\angle X = 65°$ and $\angle Y = 45°$. If triangle $XYZ$ is reflected across the perpendicular bisector of side $XZ$ to form triangle $X'Y'Z'$, what is the measure of $\angle Y'X'Z'$ in the reflected triangle?
   Answer: $65°$ because reflections preserve all angle measures exactly When you encounter reflection problems in geometry, remember that reflections are rigid transformations that preserve all distances, angles, and shapes exactly. The reflected figure is congruent to the original, just in a new position. First, let's find all angles in the original triangle. Since angles in a triangle sum to $180°$, we have $\angle Z = 180° - 65° - 45° = 70°$. When triangle $XYZ$ is reflected across the perpendicular bisector of side $XZ$, the triangle flips over that line. Points $X$ and $Z$ land on themselves (since they're on the perpendicular bisector), while point $Y$ moves to position $Y'$ on the opposite side. In the reflected triangle $X'Y'Z'$, vertex $X'$ corresponds to original vertex $X$. Since reflections preserve all angle measures, $\angle Y'X'Z' = \angle YXZ = 65°$. Choice A is incorrect because it gives $45°$, which was the measure of $\angle Y$ in the original triangle, not $\angle X$. Choice B is wrong because $115°$ doesn't correspond to any angle in this triangle, and the reasoning about "orientation change" is false—reflections don't change angle measures. Choice C incorrectly gives $70°$, which was $\angle Z$ in the original triangle. Choice D correctly identifies that $\angle Y'X'Z' = 65°$ because reflections preserve angle measures exactly. Study tip: Remember that all rigid transformations (reflections, rotations, translations) preserve angle measures. When solving reflection problems, identify corresponding vertices carefully, then apply the preservation property.

Start this quiz