Quadrilaterals

We are designing the logo for a new eSports team. The main logo is in the shape of the quadrilateral shown in the diagram. Look at its sides and angles, and answer:

1. a) What is the precise name of this quadrilateral?
2. b) Which group does it belong to: parallelograms, trapeziums or trapezoids? Justify your answer by looking at the parallelism of its sides.

In a technical drawing class, a student has drawn a quadrilateral and measured three of its interior angles: $110^\circ$, $70^\circ$ and $110^\circ$.

1. a) Knowing that the angle sum of any quadrilateral is always the same, how much must the fourth angle measure?
2. b) If opposite sides are equal in pairs, what shape is it?

State whether the following statements about parallelograms are true or false, briefly justifying your answer:

1. Every square is also a rectangle because it has 4 right angles.

2. A rhombus is a square because its 4 sides are equal.

3. A rhomboid does not have equal diagonals.

In a graphic design workshop, Marcos is creating a logo in the shape of a rhomboid (non-regular parallelogram). If one of the interior angles of the rhomboid measures $65^\circ$, calculate with clear reasoning the size of the other three angles of the logo.

In a parallelogram, the sum of two of its opposite angles is $130^\circ$.

1. How much does each of these two angles measure?

2. How much do the remaining two angles of the parallelogram measure?

3. If the longer side of the parallelogram measures $12 \text{ cm}$ and the perimetre is $38 \text{ cm}$, how long is the shorter side?

Draw in your notebook a square with a side of $4\text{ cm}$, like the one in the figure.

1. a) Draw all its axes of symmetry. How many have you found in total?
2. b) Do all the axes intersect at the same point? What is that point called?

You can do it on paper and send a photo with your answer.

Look at the following statements about quadrilaterals and state whether they are True or False, justifying your answer:

1. "A square is, at the same time, a rectangle and a rhombus."

2. "A right-angled trapezium has exactly four right angles."

3. "In a rhomboid, the diagonals are equal and perpendicular."

4. "Any parallelogram has two pairs of parallel sides."

The top of a classroom table is in the shape of a quadrilateral. If you look closely, it has two parallel sides of different lengths and the other two sides are equal to each other.

1. a) How is this quadrilateral classified?
2. b) Is it a parallelogram? Explain why.

The cross section of a new skate ramp is in the shape of an isosceles trapezium. We know that the top base measures $3\text{ m}$, the bottom base measures $7\text{ m}$ and the angles formed by the bottom base with the side edges are $72^\circ$. State the sizes of the two top angles and justify your answer using the properties of trapeziums.

In building a skatepark ramp, a structure shaped like a right-angled trapezium is used. The obtuse angle of the structure measures $125^\circ$. Calculate the measure of all the interior angles of this trapezium and justify your answer based on the properties of parallel lines cut by a transversal.

In a geometry exam, Lucía is asked to calculate the missing angle of a paper kite (a quadrilateral with two pairs of equal adjacent sides). The unequal angles measure $110^\circ$ (top) and $40^\circ$ (bottom). If the other two angles are equal to each other, how much does each of them measure? Remember to use the angle sum property of the interior angles of a quadrilateral.

In a theme park, they have designed a new play area shaped like a kite. We know that the top angle measures $\hat{A} = 110^\circ$ and the bottom angle measures $\hat{C} = 40^\circ$. The other two side angles are equal to each other. Calculate the measure of these two side angles and explain with reasons why they must be equal in this type of shape.

Look at the diagonals of the following quadrilaterals. One of the most important properties is whether they are perpendicular (forming $90^\circ$) and whether they intersect at their midpoint.

Identify which of these diagrams represents the diagonals of a rhombus and which represents those of a rectangle. Justify your choice.

Classify the following quadrilaterals based exclusively on the description of their diagonals:

• Shape A: Its diagonals are of different lengths, are perpendicular and intersect at their midpoint.

• Shape B: Its diagonals are equal, are not perpendicular and intersect at their midpoint.

• Shape C: Its diagonals are equal, are perpendicular and intersect at their midpoint.

Work out which quadrilateral is being described in each case, based on its diagonals:

1. "My diagonals are equal and intersect perpendicularly at their midpoint."

2. "My diagonals are equal, intersect at their midpoint, but are not perpendicular."

3. "My diagonals are not equal, but they are perpendicular and intersect at their midpoint."

4. "I have two parallel sides and two equal diagonals, but they do not intersect at their midpoints."

Justify your answer by making a small sketch of the diagonals for each case.

The screen of a new smartphone model is rectangular. The manufacturer states that its base measures $7\text{ cm}$ and its height is $15\text{ cm}$.

1. a) Calculate the perimetre of the smartphone (the length of the metal frame surrounding the screen).
2. b) If we want to fit a screen protector that covers the entire surface, what will the area of the protector be in $\text{cm}^2$?

A phone company wants to design a screen for a new smartphone model in the shape of a rectangle. The designer has established that the base must measure twice the height. If the total perimetre of the screen must be $36\text{ cm}$, what will the dimensions (base and height) of the screen be?

The screen of a new smartphone model is rectangular in shape. The manufacturer states that the total perimeter of the screen is $48 \text{ cm}$ and that the base is exactly twice the height.

1. Calculate the dimensions (base and height) of the screen.

2. If a protector costing $\text{RD\$}\ 0.50$ per $\text{cm}^2$ of surface area is to be fitted, how much will it cost to protect the entire screen?

Calculate the area of the shaded region on the following grid, knowing that each small square has a side length of $1\text{ cm}$. To make it easier, split the shape into a rectangle and a triangle.

A square garden with a side length of $20 \text{ meters}$ has a square fountain in its center. We know that the distance from any side of the garden to the parallel side of the fountain is $6 \text{ meters}$.

1. What is the side length of the central fountain?

2. Calculate the area of the walking area (the region of the garden not occupied by the fountain).

An architect is designing a rhombus-shaped window. To ensure structural stability, they have placed two wooden slats matching the diagonals of the rhombus. The longer diagonal measures $160\text{ cm}$ and the shorter diagonal measures $120\text{ cm}$. When crossing, the diagonals divide the rhombus into four identical triangles.

1. What type of triangles are these? (classified by their angles).

2. If the side length of the rhombus is $100\text{ cm}$, calculate the total perimetre of the window.

An artist is designing a mosaic with rhombus-shaped tiles. They know that the two diagonals of the rhombus add up to $28 \text{ cm}$ and that the longer diagonal is $4 \text{ cm}$ longer than the shorter diagonal.

1. Calculate the lengths of both diagonals.

2. Calculate the area of a single tile of the mosaic.

3. If the entire mosaic uses $50$ identical tiles, how many square decimeters ($\text{dm}^2$) of surface area will it cover in total?

Draw any quadrilateral (irregular quadrilateral) with no parallel sides in your notebook. Find the midpoint of each of its four sides and connect them following the order around the perimeter.

1. What special geometric shape is always formed inside?

2. If the sides of the original quadrilateral measure $8\text{ cm}$, $10\text{ cm}$, $12\text{ cm}$ and $14\text{ cm}$, what is the perimetre of the original quadrilateral?

You can do it on paper and send a photo with your answer.

A graphic design company is creating a logo. They have drawn a general quadrilateral and marked the midpoints of each of its sides. By joining these midpoints in order, a new quadrilateral is formed inside.

1. What type of quadrilateral is always formed inside, regardless of the shape of the original?

2. If the area of the outer quadrilateral is $120 \text{ cm}^2$, what will be the area of the shape formed by joining the midpoints?