Quadrilaterals

A quadrilateral can be split into how many triangles by drawing diagonals from a single vertex?

Identify the pair of adjacent angles in the quadrilateral $WXYZ$.

Which of the following figures is NOT a quadrilateral because its points are not joined in a simple closed loop of four segments?

A diagonal of a quadrilateral is a line segment joining two non-consecutive vertices. In the figure, which segment is a diagonal?

Identify the pair of opposite angles in the quadrilateral shown below.

Based on the definition of a convex quadrilateral, if the diagonals of quadrilateral $ABCD$ are $AC$ and $BD$, which statement must be true?

If a quadrilateral is named $EFGH$, which of the following is a pair of opposite sides?

Given a planar quadrilateral $WXYZ$, if the measure of $\angle W$ is $x$, $\angle X$ is $2x$, $\angle Y$ is $3x$, and $\angle Z$ is $4x$, determine the value of $x$ and classify the quadrilateral as convex or non-convex.

A quadrilateral is defined by vertices $A, B, C, D$ in that order. If the side $AB$ is parallel to $CD$ and the side $BC$ is parallel to $AD$, which of the following is a pair of 'adjacent sides' and a pair of 'opposite angles'?

In the study of tilings, it is noted that any quadrilateral (even irregular ones) can tile a plane. If an irregular quadrilateral $ABCD$ has angles $70^\circ, 80^\circ, 100^\circ, 110^\circ$, and four copies of this quadrilateral meet at a single vertex in a tiling, which set of angles must be present at that vertex to fill the space without gaps or overlaps?

Which property correctly describes a rhombus (a special parallelogram)?

If the perimeter of a parallelogram is $40\text{ cm}$ and one side is $12\text{ cm}$, find the length of the adjacent side.

In the rectangle $PQRS$ (which is a type of parallelogram), if $PQ = 8$ cm and $QR = 6$ cm, find the length of the diagonal $PR$.

In a parallelogram, if the ratio of two adjacent angles is $2:3$, find the measure of the smaller angle.

Two adjacent sides of a parallelogram are in the ratio $4:5$ and its perimeter is $72\text{ cm}$. Find the length of the longer side.

In the parallelogram $ABCD$, diagonal $AC$ bisects $\angle A$. If $\angle BAC = 35^\circ$, find the measure of $\angle ADC$.

Find the measure of all angles of a parallelogram if one of its angles is $30^\circ$ less than twice the smallest angle.

In a logistics warehouse, a sorting area is marked by a parallelogram $PQRS$. A robotic sensor is placed at the intersection of the diagonals $O$. Four conveyor belts are placed such that they connect points $W, X, Y, Z$ on the sides $PQ, QR, RS, SP$ respectively. If $W, X, Y, Z$ are chosen such that the segments $WY$ and $XZ$ both pass through the sensor at $O$, and $PW = RY$, which of the following statements must be true about the quadrilateral $WXYZ$?

A landscape architect designs a park in the shape of a parallelogram $ABCD$. A paved walking path is constructed along the diagonal $AC$. Two additional paths are built from vertices $B$ and $D$, perpendicular to the path $AC$, meeting it at points $X$ and $Y$ respectively. If the total length of the diagonal path $AC$ is $120\text{ m}$ and the segment $AX$ is $35\text{ m}$, what is the length of the segment $CY$?

A plot of land is in the shape of a parallelogram $ABCD$. A straight fence is built from $A$ to the midpoint $M$ of $CD$. If the area of the triangle $ADM$ is $250\text{ m}^2$, what is the total area of the land $ABCD$?

If the diagonals of a parallelogram are perpendicular to each other and equal in length, then the parallelogram is a:

The perimeter of a parallelogram is $30$ cm. If the longer side is $9$ cm, find the length of the shorter side.

In a square $ABCD$, what is the measure of $\angle CAD$?

Which of the following properties is NOT always true for a parallelogram?

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

In $\triangle ABC$, $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively. If $BC = 10$ cm, find the length of $DE$.

In a parallelogram $ABCD$, diagonal $AC$ bisects $\angle A$. What type of quadrilateral is $ABCD$?

In $\triangle ABC$, $D, E, F$ are the midpoints of sides $BC, CA, AB$ respectively. If the area of $\triangle ABC$ is $80$ cm$^2$, find the area of the parallelogram $BDEF$.

A courtyard is in the shape of a rhombus $ABCD$. The diagonals $AC$ and $BD$ have lengths 16 m and 12 m respectively. A path is created by joining the midpoints of the four sides. What is the area of the region enclosed by this path?

In a triangle $ABC$, $AD$ is the median to the side $BC$. $E$ is the midpoint of $AD$. $BE$ is produced to meet $AC$ at $F$. Using the properties of parallelograms and the Midpoint Theorem, find the ratio of $AF$ to $AC$.

In $\triangle XYZ$, $P$ is the midpoint of $XY$ and $PQ \parallel YZ$ meeting $XZ$ at $Q$. If $XZ = 12\text{ cm}$, find $XQ$.

In $\triangle ABC$, $LM$ is a line segment joining midpoints of $AB$ and $AC$. If $\angle ABC = 60^\circ$, find $\angle ALM$.

The perimeter of $\triangle ABC$ is $24\text{ cm}$. If $D, E, F$ are the midpoints of the sides $AB, BC, CA$ respectively, find the perimeter of $\triangle DEF$.

In the given figure, $E$ and $F$ are midpoints of sides $AB$ and $AC$ of $\triangle ABC$. Which of the following relations is always true according to the Midpoint Theorem?

If $D, E, F$ are midpoints of sides $AB, BC, CA$ of $\triangle ABC$, then $\triangle ABC$ is divided into four triangles which are all:

In $\triangle ABC$, the midpoints of $AB, BC$ and $CA$ are $(3,4), (8,4)$ and $(5,8)$ respectively. What is the length of side $BC$?

In the figure, $ABCD$ is a trapezium with $AB \parallel DC$. $E$ is the midpoint of $AD$ and a line through $E$ parallel to $AB$ meets $BC$ at $F$. If $AB = 8\text{ cm}$ and $DC = 12\text{ cm}$, find $EF$.

In a right-angled triangle $ABC$ (right-angled at $B$), $D$ and $E$ are the midpoints of $AB$ and $AC$. If $AB = 6$ cm and $BC = 8$ cm, find the length of the segment $DE$.

A kite-shaped plot has diagonals of lengths 16 m and 30 m. A garden is made by joining the midpoints of the four sides of the plot. What is the area of this garden?

In $\triangle XYZ$, the midpoints of $XY, YZ,$ and $ZX$ are $L, M,$ and $N$ respectively. If $XY = 8$ cm and $YZ = 10$ cm, and the perimeter of triangle $LMN$ is 12 cm, find the length of the side $XZ$.

If the coordinates of the vertices of a triangle are $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the coordinates of the centroid are:

In $\triangle ABC$, $G$ is the centroid. The area of $\triangle GBC$ is what fraction of the area of $\triangle ABC$?

A median of a triangle divides it into two triangles of:

In $\triangle PQR$, $S$ is the midpoint of $QR$. If $G$ is the centroid and $GS = 4\text{ units}$, what is the length of the median $PS$?

In $\triangle ABC$, $G$ is the centroid. If the distance from $G$ to the side $BC$ is $2\text{ cm}$, what is the height of the triangle $ABC$ from vertex $A$ to the side $BC$?

In $\triangle ABC$, the medians $AD$, $BE$, and $CF$ meet at $G$. If the sum of the squares of the sides $AB^2 + BC^2 + CA^2 = 120$, what is the sum of the squares of the medians $AD^2 + BE^2 + CF^2$?

In $\triangle ABC$, $AD$ is a median and $G$ is the centroid. If the area of $\triangle AGD$ is $5\text{ sq units}$, find the area of $\triangle ABD$.

In a triangle $ABC$, $AD$ is a median and $G$ is the centroid. A line is drawn through $G$ parallel to $BC$ intersecting $AB$ at $E$ and $AC$ at $F$. If the length of $BC$ is $12$ cm, calculate the length of the segment $EF$ using the properties of the centroid and similar triangles.

A triangular field $LMN$ has medians $LP, MQ, NR$ which intersect at the centroid $G$. If the sum of the lengths of the medians is $45$ m, what is the sum of the distances from the centroid to the three vertices ($LG + MG + NG$)?

In $\triangle XYZ$, the medians $XN$ and $YM$ intersect at the centroid $G$. If the area of $\triangle XGY$ is $24$ cm$^2$, determine the area of the quadrilateral $MZNG$.

Which of the following is NOT a property of the Varignon parallelogram $PQRS$ formed from quadrilateral $ABCD$?

For any quadrilateral $ABCD$, if $PQRS$ is the Varignon parallelogram, which of the following is true regarding its diagonals?

In a rhombus $ABCD$, $P, Q, R, S$ are the midpoints of sides $AB, BC, CD, DA$. The quadrilateral $PQRS$ is always a:

In the given diagram, $P$ and $Q$ are the midpoints of $AB$ and $BC$ of quadrilateral $ABCD$. If $AC = 10$ cm, what is the length of $PQ$?

In the figure, $ABCD$ is a quadrilateral. $P$ and $R$ are midpoints of $AB$ and $CD$. $Q$ and $S$ are midpoints of $BC$ and $DA$. If $PQRS$ is a rectangle, which must be true about $ABCD$?

If $PQRS$ is the quadrilateral formed by joining the midpoints of the sides of quadrilateral $ABCD$, and the area of $PQRS$ is $20$ units$^2$, what is the area of $\triangle ABC + \triangle ADC$?

In quadrilateral $ABCD$, $E, F, G, H$ are midpoints of $AB, BC, CD, DA$. If $AC ⊥ BD$ and $AC = BD$, then $EFGH$ is a:

In a structural layout, quadrilateral $ABCD$ has diagonals $AC$ and $BD$. Points $M_1, M_2, M_3, M_4$ are the midpoints of $AB, BC, CD, DA$. A secondary frame is created by joining the midpoints of the sides of $M_1M_2M_3M_4$, forming a third quadrilateral $K_1K_2K_3K_4$. If the perimeter of the Varignon parallelogram $M_1M_2M_3M_4$ is $60 \text{ cm}$ and $AC = 1.5 \times BD$, find the length of the diagonal $AC$.

An architect is modeling a 'dart-shaped' non-convex quadrilateral $ABCD$. The vertices are $A(0, 10)$, $B(8, 0)$, $C(0, 4)$, and $D(-8, 0)$. She marks the midpoints of sides $AB, BC, CD, DA$ as $E, F, G, H$ respectively. If a decorative glass pane is to be fitted exactly within the boundary $EFGH$, calculate the area of this glass pane.

A civil engineer designs a quadrilateral suspension frame $ABCD$. He connects the midpoints $P, Q, R, S$ of the sides $AB, BC, CD, DA$ respectively to form a central brace. The diagonals of the outer frame intersect at $O$. If the total area of the triangles at the corners (the regions $ABCD$ excluding the brace $PQRS$) is $350 \text{ cm}^2$, what is the area of the central brace $PQRS$?

A student is placing 9 identical copies of an irregular quadrilateral into a $3 \times 3$ grid of Varignon parallelograms. After placing these initial 'green' copies in the shaded cells, the student notices that 'purple' copies are automatically formed in the remaining gaps. How many such gaps (purple tiles) are completely enclosed by the green tiles in a $3 \times 3$ arrangement?

Which of the following describes why any quadrilateral (even an irregular or non-convex one) can successfully tile the plane without gaps or overlaps?

In 2023, mathematicians discovered a 13-sided shape known as the 'hat' that tiles the plane aperiodically. What does 'aperiodic' mean in the context of tiling?

If an irregular quadrilateral has an area of $24\text{ cm}^2$, what is the area of its corresponding Varignon parallelogram used in the grid-based tiling method?

In the discovery of aperiodic tilings, the 'hat' shape is a 'polykite' made of 8 smaller kites. If each kite has an area of $X$ square units, what is the total area of the minimal repeating unit in an aperiodic tiling of 'hats'?

A non-convex quadrilateral (like 'DART') has interior angles $a, b, c,$ and $d$ (where $d$ is the reflex angle). To successfully tile the plane using Method 1, what must be true about the sum of $a, b, c$ and $d$?

Method 2 of tiling uses the Varignon parallelogram grid. If the area of an irregular quadrilateral $ABCD$ is $50$ square units, what is the area of each individual parallelogram in the underlying grid used to place the tiles?

A pattern is generated by repeating a kite-shaped quadrilateral with sides $a=5$ and $b=12$, where the angle between the two unequal sides is $90^{\circ}$. To tile the plane, four such kites are arranged around a vertex such that the $90^{\circ}$ angles meet. What is the total area around this vertex covered by these four tiles?

When tiling with a general quadrilateral, we can create a central point where four different vertices $(A, B, C, D)$ meet. If the quadrilateral has vertices at $(0,0), (4,0), (3,2),$ and $(1,2)$ in a coordinate plane, and we rotate this tile $180^{\circ}$ about the midpoint of the side connecting $(3,2)$ and $(4,0)$, what are the coordinates of the new vertex corresponding to the original origin $(0,0)$?

A tiling is made of congruent quadrilaterals where each tile is a trapezoid with parallel sides of $6$ cm and $10$ cm, and a height of $4$ cm. If the tiling is formed by reflecting the trapezoid across its non-parallel sides, what is the area of a $2 \times 2$ block of these tiles?