Geometry Notes — Kites — Sketchpad Slides

Kite

Geometry Notes — Kite

A kite is a member of the quadrilateral family, and while easy to understand visually, is a little tricky to define in precise mathematical terms.

A kite is a convex quadrilateral with two adjacent sides the same length and the other two adjacent sides the same length.

Things I know about kites…

• 1. 4 sides and 4 angles
  2. the sum of the interior angle is 360 and the sum of the exterior angles is 360

• 3. Two pairs of adjacent sides of a kite are equal in length
• 4. One pair of opposite angles (the ones that are between the sides of unequal length) are congruent.
• 5. One diagonal bisects the other. They do not bisect each other, however.
• 6. Diagonals are perpendicular.
• 7. One diagonal divides a kite into two isosceles triangles; the other (the axis of symmetry) divides the kite into two congruent triangles.

A kite can become a rhombus. In the special case where all 4 sides are the same length, the kite satisfies the definition of a rhombus. If this happens, the figure doesn’t stop being a kite. But a better classification would be rhombus, which places it in the parallelogram family of quadrilaterals.  And as you know, a rhombus can become a square if its interior angles are 90°.

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Dart – Concave quadrilateral with two sets of congruent adjacent sides

Beam me up Scotty.

Pretty stuff.

Geometry Notes — Trapezoid, Isosceles Trapezoid — Sketchpad Slides

Trapezoid

Isosceles Trapezoid

Geometry Notes — Trapezoid, Isosceles Trapezoid, Median of a Trapezoid

Trapezoid – a quadrilateral with exactly one pair of parallel sides

Base – One of the parallel sides. Every trapezoid has two bases.
Leg – The non-parallel sides are legs. Every trapezoid has two legs.
Altitude – The altitude of a trapezoid is the perpendicular distance from one base to the other. (One base may need to be extended).
Median – The median of a trapezoid is a segment joining the midpoints of the two legs.

Things I know about trapezoids

• 1.  4 sides
• 2.  sum of the interior angles is 360

• 3.  one pair of parallel sides

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If both legs are the same length, this is called an isosceles trapezoid, and both base angles are the same.

If the legs are parallel, it now has two pairs of parallel sides, and is a parallelogram.

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Isosceles trapezoid – a trapezoid with congruent legs

Things I know about isosceles trapezoids

• 1.  4 sides
• 2.  sum of the interior angles is 360

• 3.  one pair of parallel sides

• 4.  congruent legs
• 5.  base angles congruent
• 6.  opposite angles are supplementary
• 7.  diagonals are congruent

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Midsegment or Median of a Trapezoid – a segment that connects the midpoints of a trapezoid’s legs

The median of a trapezoid is

• parallel to both bases
• has length equal to the average of the length of the bases

In both examples above, AB and CD are parallel and the median is equal to one-half the sum of the two bases, meaning the median = 1/2(AB + CD)

I love geometry art. Outstanding.

Geometry Notes — Rhombus, Rectangle, Square — Sketchpad Slides

Rhombus

Rectangle

Square

Geometry Notes — Special Parallelogram — Square

Square – a parallelogram with four right angles and all sides congruent

The square is probably the best known of the quadrilaterals.

A square is a regular polygon.

Thing I know about Squares

• 1. 4 sides and 4 angles
• 2. the sum of the interior angles is 360 and the sum of the exterior angles is 360

• 3.  opposite sides are parallel
• 4.  opposite sides are congruent
• 5.  opposite angles are congruent
• 6.  consecutive angles are supplementary
• 7.  diagonals bisect each other
• 8.  a single diagonal will divide a square into two congruent triangles

• 9.  four right angles   or   all angles congruent
• 10.  all sides congruent
• 11.  diagonals are congruent
• 12.  diagonals are perpendicular
• 13.  diagonals bisect opposite angles

A square can be thought of as a special case of other quadrilaterals, for example

• a rectangle but with adjacent sides equal
• a parallelogram but with adjacent sides equal and the angles all 90°
• a rhombus but with angles all 90°

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Pretty square art.

Geometry Notes — Special Parallelogram — Rhombus

Rhombus – a parallelogram with all sides congruent

A rhombus is a special type of parallelogram. With a rhombus, all four sides are the same length. Its a bit like a square that can ‘lean over’ and the interior angles need not be 90°. Sometimes called a diamond shape. That is how I would draw it — exaggerate the diamond shape.

Things I know about a rhombus

• 1.  4 sides and 4 angles
• 2.  the sum of the interior angles is 360 and the sum of the exterior angles is 360

• 3.  opposite sides are parallel
• 4.  opposite sides are congruent
• 5.  opposite angles are congruent
• 6.  consecutive angles are supplementary
• 7.  diagonals bisect each other
• 8.  a single diagonal will divide a rhombus into two congruent triangles

• 9.  all sides congruent
• 10.  diagonals are perpendicular
• 11.  diagonals bisect opposite angles

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Geometry Notes — Special Parallelogram — Rectangle

Rectangle – a parallelogram with four right angles

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).

Things I know about a rectangle

• 1.  4 sides and 4 angles
• 2.  the sum of the interior angles is 360 and the sum of the exterior angles is 360

• 3.  opposite sides are parallel
• 4.  opposite sides are congruent
• 5.  opposite angles are congruent
• 6.  consecutive angles are supplementary
• 7.  diagonals bisect each other
• 8.  a single diagonal will divide a rectangle into two congruent triangles

• 9.  four right angles   or  all angles are congruent
• 10.  diagonals are congruent

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Pretty. Geometric Art.  Who knew? Me.

 

Geometry Notes — Proving that Quadrilaterals are Parallelograms

Proving that Quadrilaterals are Parallelograms

1.  by definition… If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.

2.  If the opposite sides of a quadrilateral are congruent, then it is a parallelogram.

3.  If the opposite angles of a quadrilateral are congruent, then it is a parallelogram.

4.  If an angle of a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram.

5.  If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

6.  If one pair of opposite sides of a quadrilateral is both congruent and parallel, then it is a parallelogram.

Geometry — Biography — Hippasus

Hippasus of Metapontum, ca. 500 B.C., was a Greek philosopher. He was a disciple of Pythagoras. Hippasus is attributed the discovery of the existence of irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is irrational.

It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn’t believe there was a need for any other numbers.

Up until Hippasus’ discovery, the Pythagoreans preached that all numbers could be expressed as the ratio of integers. Despite the validity of his discovery, the Pythagoreans initially treated it as a kind of religious heresy and they either exiled or murdered Hippasus. Legend has it that the discovery was made at sea and that Hippasus’ fellow Pythagoreans threw him overboard.

The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. An event would happen that would change the very core of their beliefs. Along came Pythagorean Hippasus who discovered that the diagonal of a square whose side was one unit could not be expressed as a whole number or a ratio. Hence, the Pythagorean Theorem which crushed their original beliefs. Thus, they certainly didn’t want Hippasus’ discovery to be revealed and shatter their pride and core beliefs.

In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n not equal to zero. It can be deduced that they also cannot be represented as terminating or repeating decimals.

Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. 3.1415926535897932384626433832795 (and more …)

The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. 2.7182818284590452353602874713527 (and more …)

The square root of 2, also known as Pythagoras’ constant (ironic huh), is 1.41421356237309504880168872420969807856967187537694807317667973799  (and more …)