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From Wikipedia

Quadrilateral

In Euclidean plane geometry, a quadrilateral is a polygon with four sides (or 'edges') and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave.

The interior angles of a simple quadrilateral add up to 360 degrees of arc. In a crossed quadrilateral, the interior angles on either side of the crossing add up to 720°.

All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

Convex quadrilaterals - parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms also include the square, rectangle, rhombus and rhomboid.

  • Rhombus or rhomb: all four sides are of equal length. Equivalent conditions are that opposite sides are parallel and opposite angles are equal, or that the diagonals perpendicularly bisect each other. An informal description is "a pushed-over square" (including a square).
  • Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). Informally: "a pushed-over rectangle with no right angles."
  • Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other and are equal in length. Informally: "a box or oblong" (including a square).
  • Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), that the diagonals perpendicularly bisect each other, and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
  • Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e. a rectangle that is not a square).

A shape that is both a rhombus (four equal sides) and a rectangle (four equal angles) is a square (four equal sides and four equal angles).

Convex quadrilaterals - other

  • Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. (It is common, especially in the discussions on plane tessellations, to refer to the concave quadrilateral with these properties as a dart or arrowhead, with term kite being restricted to the convex shape.)
  • Orthodiagonal quadrilateral: the diagonals cross at right angles.
  • Trapezium (British English) or trapezoid (NAm.): one pair of opposite sides are parallel.
  • Isosceles trapezium (Brit.) or isosceles trapezoid (NAm.): one pair of opposite sides are parallel and the base angles are equal in measure. This implies that the other two sides are of equal length, and that the diagonals are of equal length. An alternative definition is: "a quadrilateral with an axis of symmetry bisecting one pair of opposite sides".
  • Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
  • Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A quadrilateral is cyclic if and only if opposite angles sum to 180°.
  • Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
  • Bicentric quadrilateral: both cyclic and tangential.

Area of a convex quadrilateral

There are various general formulas for the area of a convex quadrilateral.

The area of a quadrilateral ABCD can be calculated using vectors. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then

\frac{1}{2} |{AC}\times{BD}|,

which is the magnitude of the cross product of vectors AC and BD. In two-dimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x1,y1) and BD as (x2,y2), this can be rewritten as:


Cyclic quadrilateral

In geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. The vertices are said to be concyclic.

In a cyclic simple (non-self-intersecting) quadrilateral, opposite angles are supplementary (their sum is π radians or 180°). Equivalently, each exterior angle is equal to the opposite interior angle.

Area

The area of a cyclic quadrilateral is given by Brahmagupta's formula as long as the sides are given:

\sqrt{(s-a)(s-b)(s-c)(s-d)}

where s, the semiperimeter, is s=\frac{a+b+c+d}{2}.

This area is maximal among all quadrilaterals having the same sequence of side lengths.

The area of a cyclic quadrilateral with successive sides a, b, c, d and angle \gamma between sides b and c can also be expressed as

Area=\tfrac{1}{2}(\sin \gamma)(bc+ad).

Diagonals

Ptolemy's theorem expresses the product of the lengths of the two diagonals p and q of a cyclic quadrilateral as equal to the sum of the products ac and bd of opposite sides:

pq = ac + bd.

In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.

A cyclic quadrilateral with successive vertices A, B, C, D and successive sides a=AB, b=BC, c=CD, and d=DA and with diagonals p=AC and q=BD has:

\tfrac {p}{q}= \tfrac{ad+cb}{ab+cd},
p^{2}= \tfrac{(ac+bd)(ad+bc)}{ab+cd},

and

q^{2}= \tfrac{(ac+bd)(ab+dc)}{ad+bc}.

If the intersection of the diagonals divides one diagonal into segments of lengths e and f, and divides the other diagonal into segments of lengths g and h, then ef = gh. (This holds because both diagonals are chords of a circle.)

Special cases

Any square, rectangle, or isosceles trapezoid is cyclic. A kite is cyclic if and only if it has two right angles.

Other properties

A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s has circumradius (the radius of the circumscribing circle) given by

\tfrac{1}{4} \sqrt{\tfrac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.

There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression.

For a cyclic quadrilateral with successive sides a, b, c, d, semiperimeter s, and angle A between sides a and d, the trigonometric functions of A are given by

\cos A = \frac{a^2 + d^2 - b^2 - c^2}{2(ad + bc)};
\sin^2 A = \frac{4(s-a)(s-b)(s-c)(s-d)}{(ad+bc)^2};
\tan \frac{A}{2} = \sqrt{\frac{(s-a)(s-d)}{(s-b)(s-c)}}.

Four lines, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.

Properties of cyclic quadrilaterals that are also orthodiagonal

Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal (has mutually perpendicular diagonals), the perpendicular from any side through the point of intersection of the diagonals bisects the other side.

If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the length of the opposite side.

For a cyclic orthodiagonal quadrilateral, suppose the intersection of the diagonals divides one diagonal into segments of lengths p1 and p2 and divides the other diagonal into segments of lengths q1 and q2. Then

where D is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. Equivalently, letting R = D / 2 be the radius of the circumcircle, the average of p_1^2, p_2^2, q_1^2, and q_2^2 is R^2. Moreover, the equations a2 + c2 = b2 + d2 = D2 imply that in an orthodiagonal cyclic quadrilateral, the sum of the squares of the sides equals eight times the square of the circumradius.



From Yahoo Answers

Question:thankyou hunnies x

Answers:Threre isn't a specific name for a quadrilateral with only one right angle. However, if the questions means at least one, then a square or rectangle would fit the bill, as well as some trapezoids.

Question:The Vertices of a quadrilateral prove the diagonals intersect at right angles? The Vertices of a quadrilateral are A(1,4), B(5,11), C(10,10), and D(11,2). Prove the diagonals intersect at right angles. How would i go about doing this? Thanks for any help in advance

Answers:1) For the given quadrilateral, ABCD, taken in order, AC is one diagonal and BD is another diagonal. 2) Slope AC = (10 - 4)/(10 - 1) = 6/9 = 2/3 3) Slope BD = (2 - 11)/(11 - 5) = -9/6 = -3/2 4) ==> Product of their slopes = 2/3 x -3/2 = -1 As product of the slopes is -1, the diagonals intersect at right angles.

Question:If a quadrilateral has 4 congruent sides, but its diagnols are not congruent, then the quadrilateral is a ...??? (diagonal- lines drawn from 1 vertex 2 another.) Which description of a parallelogram is incorrect?? a) quadrilateral with 2 pairs of parallel opposite sides b) quadrilateral with opposite sides congruent c) 4-sided polygon with opposite sides parallel d) polygon with opposite sides parallel

Answers:First question: RHOMBUS. A rhombus has four congruent sides, but all four angles are not congruent (although opposite angles are). Since the angles are not all congruent, the diagonals are not congruent. Second question: D. A parallelogram has 4 sides. It does not describe any polygon.

Question:And why is this not possible with 3 right angles? EXPALIN PLEASE. or is it possible?

Answers:It IS possible to draw a quadrilateral that has 2 right angles but is not a rectangle. Draw a trapezium with two adjacent right angles. If there are 3 right angles, then the fact that the sum of the interior angles of a quadrilateral is 360 forces the fourth angle to be 90 . A quadrilateral can have 0, 1, 2 or 4 right angles. I hope this helps.

From Youtube

Quadrilaterals and Right Angles :WEBSITE: www.teachertube.com KS2 SATs question

Sum of interior angles of a quadrilateral :Using Geometer Sketchpad 4 to determine to sum of the angles in a quadrilateral