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 .
Their vertices are marked with capital letters and sides with small letters.
Angles in vertices and are usually marked in order with: (alpha, beta, gamma, delta).
Remember that in triangles, the sum of the measures of all exterior angles is equal to (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 .
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 is an angle in vertex , in vertex , in vertex and in vertex , then in a trapezoid is valid:
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: , .
And it is also true, just like in any other trapezoid (note that a parallelogram is just a specific kind of a trapezoid): , .
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 and as shown below:
A rhombus is a parallelogram which have all sides of equal lengths.
Rhombuses have the same properties as parallelograms:
Opposite angles are of equal measure: , 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.
Square is a rectangle whose all sides are equal.
Diagonals in a square are congruent and perpendicular.
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 (square meter).
1 square meter is equal to the surface enclosed by a square with sides 1m.
There are also some derived units of measure for areas, for smaller or larger shapes.
Area of a square is equal to a square of length of its side.
Area of a rectangular is equal to the product of lengths of adjacent sides.
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.
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 = 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.
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.
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 and .
That means that
First notice some properties of drawn triangles. Triangles and 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 and .
Now, area of whole rectangle is equal to (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.
We insert that in our equation and get:
we can divide this equation with 2 and get:
and since this means .
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 , on side a and on side b
In words, area of a triangle is equal to the product of lengths of a side and altitude to that side.
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) =
Area of a trapezoid is equal to one half of a product of sum of its bases and altitude.
This formula is a result of dividing a trapezoid into a two triangles and , and a rectangle .
Now, we can write our area as the sum of smaller areas: .
We know that .
now we need to find and . If we translate side b next to we get a triangle .
The altitude of a triangle is equal to the altitude of a trapezoid .
And side on which this altitude is set is equal to . This leads to a conclusion that
This means that
Naming quadrilaterals (278.7 KiB, 315 hits)
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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)