PLANE FIGURES EXAM
1. RECTANGLE
![Picture](/uploads/2/2/9/3/22930296/7026019.jpg?457)
A 4-sided polygon where all interior angles are 90° Try this Drag the orange dots on each vertex to reshape the rectangle. The rectangle, like the square, is one of the most commonly known quadrilaterals. It is defined as having all four interior angles 90° (right angles).
Properties of a rectangle
http://www.mathopenref.com/rectangle.html
Properties of a rectangle
- Opposite sides are parallel and congruent Adjust the rectangle above and satisfy yourself that this is so.
- The diagonals bisect each other
- The diagonals are congruent
http://www.mathopenref.com/rectangle.html
2. SQUARE
![Picture](/uploads/2/2/9/3/22930296/1464241.jpg?331)
Square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90-degree angles, or right angles).[1] It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ABCD.
http://en.wikipedia.org/wiki/Square
- The diagonals of a square are (about 1.414) times the length of a side of the square. This value, known as Pythagoras' constant, was the first number proven to be irrational.
- A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
- If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
- If a circle is circumscribed around a square, the area of the circle is (about 1.571) times the area of the square.
- If a circle is inscribed in the square, the area of the circle is (about 0.7854) times the area of the square.
- A square has a larger area than any other quadrilateral with the same perimeter.[5]
- A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
- The square is in two families of polytopes in two dimensions: hypercube and the cross polytopes. The Schläfli symbol for the square is {4}.
- The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D4.
- If the inscribed circle of a square ABCD has tangency points E on AB, F on BC, G on CD, and H on DA, then for any point P on the inscribed circle,
http://en.wikipedia.org/wiki/Square
3. RHOMBUS
![Picture](/uploads/2/2/9/3/22930296/2520468.jpg?439)
"Rhombus" comes from the Greek ῥόμβος (rhombos), meaning something that spins which derives from the verb ρέμβω (rhembō), meaning "to turn round and round". The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for two right circular cones sharing a common base.
Characterizations
A simple (non self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:
Characterizations
A simple (non self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:
- a quadrilateral with four sides of equal length (by definition)
- a quadrilateral in which the diagonals are perpendicular and bisect each other
- a quadrilateral in which each diagonal bisects two opposite interior angles
- a parallelogram in which at least two consecutive sides are equal in length
- a parallelogram in which the diagonals are perpendicular
- a parallelogram in which a diagonal bisects an interior angle
4. TRAPEZOID
![Picture](/uploads/2/2/9/3/22930296/1146388.jpg?474)
Trapezoid is a type of plane figure which has four sides and four vertices. It has a pair of parallel side. The parallel sides are called as the base.The other two sides are called as the legs and the lateral side.
Characterizations Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
Characterizations Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
- It has two adjacent angles that are supplementary, that is, they add up 180 degrees.
- The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
- The diagonals cut each other in mutually the same ratio (this ratio is the same as that between the lengths of the parallel sides).
- The diagonals cut the quadrilateral into four triangles of which one opposite pair are similar.
- The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas
- The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.
- The areas S and T of some two opposite triangles of the four triangles formed by the diagonals verify the equation
- The midpoints of two opposite sides and the intersection of the diagonals are collinear.
- The consecutive sides a, c, b, d and the diagonals p, q verify the equation
- The distance v between the midpoints of the diagonals verifies the equation
http://en.wikipedia.org/wiki/Trapezoid
5. CIRCLE
![Picture](/uploads/2/2/9/3/22930296/4444093.jpg?465)
A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.
Terminology
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.
A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.
Terminology
- Arc: any connected part of the circle.
- Centre: the point equidistant from the points on the circle.
- Chord: a line segment whose endpoints lie on the circle.
- Circumference: the length of one circuit along the circle.
- Diameter: the longest chord, a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a segment, which is the largest distance between any two points on the circle.
- Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
- Secant: an extended chord, a straight line cutting the circle at two points.
- Sector: a region bounded by two radii and an arc lying between the radii.
- Segment: a region bounded by a chord and an arc lying between the chord's endpoints.
- Semicircle: a region bounded by a diameter and an arc lying between the diameter's endpoints. It is a special case of a segment.
- Tangent: a straight line that touches the circle at a single point.
6. PENTAGON
![Picture](/uploads/2/2/9/3/22930296/8545011.jpg?383)
Pentagon (from pente, which is Greek for the number 5) is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.
Regular pentagons In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). Its Schläfli symbol is {5}. The diagonals of a regular pentagon are in golden ratio to its sides.
The area of a regular convex pentagon with side length t is given by
A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression
Derivation of the area formula The area of any regular polygon is:
where P is the perimeter of the polygon, a is the apothem. One can then substitute the respective values for P and a, which makes the formula:
with t as the given side length. Then we can then rearrange the formula as:
and then, we combine the two terms to get the final formula, which is:
en.Wikipedia.org/wiki/The_Pentagon
Regular pentagons In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). Its Schläfli symbol is {5}. The diagonals of a regular pentagon are in golden ratio to its sides.
The area of a regular convex pentagon with side length t is given by
A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
When a regular pentagon is inscribed in a circle with radius R, its edge length t is given by the expression
Derivation of the area formula The area of any regular polygon is:
where P is the perimeter of the polygon, a is the apothem. One can then substitute the respective values for P and a, which makes the formula:
with t as the given side length. Then we can then rearrange the formula as:
and then, we combine the two terms to get the final formula, which is:
en.Wikipedia.org/wiki/The_Pentagon