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What are the types of quadrilaterals?

Quadrilaterals are part of a plane enclosed by four side (quad means four and lateral means side). All quadrilaterals have exactly four sides and four angles, and they can be sorted into specific groups based on lengths of their sides or measures of their angles. What they have in common is that in every quadrilateral the sum of the measures of all interior angles is equal to 360^{\circ}.

Their vertices are marked with capital letters and sides with small letters.

Angles in vertices A, B, C and D are usually marked in order with: \alpha, \beta, \gamma, \delta (alpha, beta, gamma, delta).

Remember that in triangles, the sum of  the measures of all exterior angles is equal to 360^{\circ} (remember, exterior angle is a supplementary angle to a certain interior angle).

This will also be true for quadrilaterals. The sum of all measures of exterior angles in quadrilaterals is always equal to 360^{\circ}.

Diagonals are lines that connect opposite angles.

First group of quadrilaterals is a scalene quadrilateral. Scalene quadrilateral is a quadrilateral that doesn’t have any special properties; his sides have different length and angles have different measure.

Quadrilaterals which have one pair of parallel sides are called trapezoids. Sides that are parallel are called bases of a trapezoid, and ones that are not parallel are called legs.

Trapezoids whose legs are of equal length are called isosceles trapezoids.

Diagonals of a isosceles trapezoids are congruent.

Height or altitude of a trapezoid is the length of a line that is perpendicular to a base and goes through opposite vertex. Altitude of a trapezoid will be equal no matter from which vertex you draw it. If you’re drawing an altitude from larger base, you simply extend the shorter base.

If \alpha is an angle in vertex A, \beta in vertex B, \gamma in vertex C and \delta in vertex D, then in a trapezoid ABCD is valid:

\alpha + \delta = 180^{\circ}
\beta + \gamma = 180^{\circ}.

This means that angles which are enclosed by a certain leg and bases are supplementary.

A parallelogram is a quadrilateral whose opposite sides are of equal length and parallel. Opposite angles of a parallelogram are of equal measure: \alpha = \gamma, \beta = \delta.

And it is also true, just like in any other trapezoid (note that a parallelogram is just a specific kind of a trapezoid): \alpha + \delta = 180^{\circ}, \beta + \gamma = 180^{\circ}.

Altitude or a height of a parallelogram is defined just like in a trapezoid: The line segment that connects a vertex with opposite side, and is perpendicular to that side.

Diagonals of a parallelogram bisect each other, which means that |AS|=|CS|=\frac{e}{2} and |BS|=|DS|=\frac{f}{2} as shown below:

rhombus is a parallelogram which have all sides of equal lengths.

rhombus

Rhombuses have the same properties as parallelograms:
Opposite angles are of equal measure: \alpha = \gamma, \beta = \delta and that adjacent angles are supplementary. The only difference between rhombuses and parallelograms is that rhombuses have all sides equal.

Diagonals in rhombus are congruent and perpendicular.

 

A rectangle is a parallelograms whose all angles are right.

Diagonals in rectangles are congruent.

rectangles

Square is a rectangle whose all sides are equal.

Diagonals in a square are congruent and perpendicular.

square

 

Perimeters and areas of triangles and quadrilaterals

Perimeter of any geometric shape is the length of is outline.

Area of any geometric shape is the surface it occupies. Unit of measure for area is m^2 (square meter).

1 square meter is equal to the surface enclosed by a square with sides 1m.

one square meter

There are also some derived units of measure for areas, for smaller or larger shapes.

1 km^2 = 1 km * 1 km = 1000 m * 1000 m = 1 000 000 m^2

1 m^2 = 1 m * 1m

1 dm^2 = \frac{1}{10} m^2 * \frac{1}{10} m^2 = \frac{1}{100} m^2

1 cm^2 = \frac{1}{100} m^2 * \frac{1}{100} m^2 = \frac{1}{10 000} m^2

1 mm^2 = \frac{1}{1000} m^2 * \frac{1}{1000} m^2 = \frac{1}{1 000 000} m^2

 

Area of a square is equal to a square of length of its side.

area of square

Area of a rectangular is equal to the product of lengths of adjacent sides.

area of rectangular

Area of a rhombus is equal to the product of length of its side and altitude. This is true because, from the picture: if we translate altitude1 into point A, and extend side FD we’ll get triangle F’DA which is congruent with triangle FCB. If we ‘translate’ triangle FCB onto triangle F’DA we’ll get a rectangular with one side a and other h.

area of rhombus

The same that goes for a rhombus works on a parallelogram, the area of a parallelogram is a product of its one side and altitude on that side.

Perimeter of a triangle is the sum of all lengths of its sides.

perimeter of triangle

Perimeter = a + b + c

For calculating area of a triangle we will use the height or altitude of a triangle. The altitude of a triangle is a line that connects a vertex to its opposite side and is perpendicular to that side.

altitude of triangle

How would we calculate the surface of this triangle? We want to try to make a shape out of this triangle to a shape we already know its surface. We’ll translate altitude from point c to the point B and A.

calculate surface of triangle

Now we got a rectangle whose sides are c and altitude, we’ll mark altitude with h (height) for easier writing and area of a certain shape A(shape).

We’ll divide area of this triangle ABC into two areas of triangles AHC and HBC.
That means that A(ABC) = A(AHC) + A(HBC)

First notice some properties of drawn triangles. Triangles AHC and IAC have two equal sides ( b is their common side, and altitude h) and one right triangle. By theorem SSA these triangles are congruent.

And you notice the same with triangles CBH and BCJ.

Now, area of whole rectangle is equal to A(ICA) + A(ACH) + A(CHB) + A(BCJ) = c * h (because area of a rectangle is equal to multiplication of lengths of its adjacent sides).

Now you look at your equation and see that since you have two pairs of congruent triangles, their area is the same.

A(ICA) = A(AHC) and A(HBC) = A(BJC)

We insert that in our equation and get:

2 A(AHC) + 2 A(HBC) = c * h we can divide this equation with 2 and get:
A(AHC)+ A(HBC) = c * \frac{h}{2} and since A(AHC)+ A(HBC) = A(ABC) this means A(ABC)= c * \frac{h}{2}.

Note that you can draw altitude from any vertex. Altitudes from different vertices have different length.

To be clear, altitude on side c will be marked h_c, on side a h_a and on side b h_b

In words, area of a triangle is equal to the product of lengths of a side and altitude to that side.

calculation area triangle

In some special cases, like the right triangle, sides a and b are overlapping with altitudes ha and hb which means that A(right triangle) = \frac{a * b}{2}

Area of a trapezoid is equal to one half of a product of sum of its bases and altitude.

area of trapezoids

This formula is a result of dividing a trapezoid into a two triangles ADG and BCF, and a rectangle DCGH.

Now, we can write our area as the sum of smaller areas: A(ABCD) = A(ADG) + A(HBC) + A(DGHC).

We know that A(DGHC) =h * c.
now we need to find A(ADG) and A(HBC). If we translate side b next to ADG we get a triangle ADE.

altitude of triangle is equal to altitude of trapezoid

The altitude of a triangle AED is equal to the altitude of a trapezoid ABCD.

And side on which this altitude is set is equal to a - c. This leads to a conclusion that

\ A (ADE) = h * \frac{a - c}{2}

This means that

A (ABCD) = A (ADE) + A (GHCD) = \frac{h * (a - c)}{2} + h_c = \frac{(h_a - h_c + 2h_c)}{2} = \frac{(h_a + h_c)}{2} = \frac{h * (a + c)}{2}.

 

Quadrilaterals worksheets

  Naming quadrilaterals (278.7 KiB, 315 hits)

  Name the biggest number of quadrilaterals (295.6 KiB, 254 hits)

  Angles in quadrilaterals (423.6 KiB, 228 hits)

  Parallelograms - Find an angle (531.7 KiB, 251 hits)

  Parallelograms - Find an length (547.3 KiB, 244 hits)

  Trapezoids - Find a length of the median (293.1 KiB, 241 hits)

  Trapezoids - Find a length of the half-segment (289.9 KiB, 227 hits)

  Trapezoids - Find a length of a base (312.8 KiB, 254 hits)

  Trapezoids - Angles (284.8 KiB, 284 hits)

  Area of triangles and quadrilaterals (501.9 KiB, 320 hits)

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