4-gons (Quadrilaterals)

If $PQRS$ is a parallelogram, then which of the following is always true regarding its diagonals $PR$ and $QS$?

True or False: Diagonals of a parallelogram are always equal in length.

In parallelogram $ABCD$, if $\angle D = 115^{\circ}$, find the measure of $\angle C$.

The ratio of two adjacent angles of a parallelogram is $2:3$. Find the measure of the smaller angle.

In the given parallelogram $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. If $AO = (3x - 5)$ cm, $OC = (x + 7)$ cm, $BO = (y + 1)$ cm, and $OD = (2y - 3)$ cm, find the values of $x$ and $y$ based on the properties of a parallelogram.

The perimeter of a parallelogram is $22$ cm. If the longer side is $6.5$ cm, find the length of the shorter side.

In parallelogram $ABCD$, the diagonals intersect at $O$. If $\angle AOB = 70^{\circ}$ and $\angle OBC = 40^{\circ}$, find $\angle OAD$.

Given a parallelogram $ABCD$ where $AB = 10$ cm and $AD = 6$ cm. If the bisector of $\angle A$ meets $DC$ at $E$, find the length of segment $EC$.

In the given figure, $ABCD$ is a parallelogram. $P$ and $Q$ are points on diagonal $BD$ such that $DP = BQ$. What is the relationship between segments $AP$ and $CQ$?

In a parallelogram $ABCD$, $\angle A$ is $30^\circ$ more than twice $\angle B$. Find the measure of the largest angle of the parallelogram.

In $\triangle PQR$, $X$ and $Y$ are points on $PQ$ and $PR$ such that $PX = XQ$ and $XY \parallel QR$. If $PR = 14$ cm, find $PY$.

In $\triangle ABC$, $L$ and $M$ are midpoints of $AB$ and $AC$. If $LM = 5$ cm, what is the length of $BC$?

In the triangle $ABC$, $D$ and $E$ are midpoints of $AB$ and $AC$. If $DE$ is parallel to $BC$, then which of the following is true?

The quadrilateral formed by joining the midpoints of the sides of a rhombus is a:

In $\triangle ABC$, $D$ and $E$ are the midpoints of sides $AB$ and $AC$ respectively. $DE$ is extended to $F$ such that $DE = EF$. If $BC = 10\text{ cm}$ and $AB = 8\text{ cm}$, find the length of the segment $CF$.

In a quadrilateral $ABCD$, if the midpoints of the sides are joined to form a square, then the original quadrilateral $ABCD$ must have:

If $ABCD$ is a rectangle and $P, Q, R, S$ are midpoints of $AB, BC, CD, DA$ respectively, then $PQRS$ is a rhombus. If $AB = 8$ and $BC = 6$, find the length of side $PQ$.

In $\triangle ABC$, the midpoint of $AB$ is $L$ and the midpoint of $AC$ is $M$. If a line is drawn from $A$ to a point $N$ on $BC$, and $P$ is the midpoint of $AN$, prove that $L, P, M$ are collinear. If $LM = 12\text{ cm}$, what is the length of $LP$ given that $L, P, M$ are points on a straight line in that order and $P$ is the midpoint of $LM$?

A metal sheet is in the shape of a quadrilateral $ABCD$. A worker cuts the sheet along the lines joining the midpoints of adjacent sides to form a smaller piece $PQRS$. If the diagonals of $ABCD$ are equal ($AC=BD$), which of the following is true about the shape of $PQRS$?

In a parallelogram $ABCD$, $E$ and $F$ are the midpoints of sides $AB$ and $CD$ respectively. The line segment $AF$ and $CE$ intersect the diagonal $BD$ at points $P$ and $Q$. If $BD = 15\text{ cm}$, find the length of $PQ$.

A square is placed on a coordinate plane with its center at $(0,0)$. If one vertex is at $(1,1)$, where is the vertex symmetric to it about the origin?

Which property of a parallelogram is directly proven by its central symmetry about the intersection of its diagonals?

If a figure has central symmetry about point $O$, then for every point $P$ in the figure, there exists a point $P'$ in the figure such that $O$ is the:

In a rectangle $ABCD$, which point serves as the center of rotational symmetry for a $180^{\circ}$ turn?

Which transformation is equivalent to a rotation of $180^{\circ}$ about the point $(2, 0)$ for a quadrilateral centered at $(2, 0)$?

A point $A(2, 3)$ is reflected in the origin to $A'$, and then $A'$ is reflected in the x-axis to $A''$. What are the coordinates of $A''$?

In a coordinate plane, a square has center $(1, 1)$ and one side parallel to the x-axis. If the length of the side is 2, what is the image of vertex $(0, 0)$ after central symmetry about the center?

The quadrilateral $ABCD$ with vertices $A(2, 2)$, $B(6, 2)$, $C(6, 6)$, and $D(2, 6)$ is transformed by a central symmetry with respect to the point $M(4, 4)$. If the resulting square is then scaled by a factor of $0.5$ with the origin as the center, what is the new coordinate of the image of point $B$?

A quadrilateral has vertices $A(1, 1)$, $B(3, 1)$, $C(4, 3)$, and $D(2, 3)$. After a rotation of $180^\circ$ about the midpoint of diagonal $AC$, what are the new coordinates of vertex $B$?

Square $S_1$ has vertices at $(\pm 1, \pm 1)$. Square $S_2$ is obtained by rotating $S_1$ by $45^\circ$ counter-clockwise about the origin. What is the area of the region common to $S_1$ and $S_2$?

Given a line segment with endpoints $A(2, -3)$ and $B(10, 5)$, what is the x-coordinate of its midpoint?

The coordinates of the midpoints of the sides of a triangle are $(1, 1)$, $(2, 3)$, and $(4, 1)$. These midpoints form a parallelogram with one vertex of the triangle. Find the midpoint of the segment connecting $(1, 1)$ and $(4, 1)$.

Find the coordinates of the fourth vertex $D$ of a parallelogram $ABCD$ if $A(1, 1)$, $B(5, 1)$, and $C(6, 4)$ are given.

In a parallelogram $PQRS$, $P = (1, 1)$, $Q = (4, 1)$, and $R = (5, 3)$. Find the midpoint of diagonal $QS$.

If $M(3, 5)$ is the midpoint of the diagonal $AC$ of parallelogram $ABCD$, and $B$ is $(2, 2)$, what are the coordinates of $D$?

A parallelogram has vertices $A(1, 1)$, $B(4, 2)$, and $C(5, 5)$. The fourth vertex $D$ is $(x, y)$. Find $x+y$.

If the vertices of a parallelogram are $A(0, 0)$, $B(a, 0)$, $C(a+b, c)$, and $D(b, c)$, what is the midpoint of the diagonal $BD$?

A rhombus is a parallelogram with all sides equal. If the vertices of a rhombus are $(3, 0)$, $(4, 5)$, $(-1, 4)$, and $(-2, -1)$, find the coordinates of the point of intersection of its diagonals.

The points $A(x, 2)$, $B(3, 4)$, and $C(7, 8)$ are three vertices of a parallelogram $ABCD$. If the intersection of diagonals is $(5, 5)$, find the value of $x$ and the coordinates of $D$.

Find the area of a parallelogram whose vertices are $A(1, 0)$, $B(4, 0)$, $C(5, 3)$, and $D(2, 3)$ by considering the distance between parallel sides.