Plane geometry quadrilaterals
58 flashcards covering Plane geometry quadrilaterals for the GMAT Quantitative section.
Plane geometry quadrilaterals are four-sided shapes that lie flat on a plane, such as squares, rectangles, parallelograms, and trapezoids. They are defined by their sides, angles, and diagonals, with properties like opposite sides being equal or parallel in certain types. Understanding these shapes helps in solving problems involving measurements, as quadrilaterals are fundamental to geometry and appear in real-world applications like architecture and design.
On the GMAT Quantitative section, quadrilaterals show up in problem-solving and data sufficiency questions that require calculating areas, perimeters, angles, or diagonals. Common traps include mistaking one quadrilateral for another, overlooking non-right angles, or misapplying formulas, so accuracy in identifying properties is key. Focus on mastering rules like the sum of interior angles equaling 360 degrees and practicing with diagrams to handle these efficiently.
Always draw a quick sketch to visualize the problem.
Terms (58)
- 01
Quadrilateral
A quadrilateral is a polygon with four sides and four angles, formed by connecting four non-collinear points.
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Convex quadrilateral
A convex quadrilateral is a four-sided polygon where all interior angles are less than 180 degrees and no sides bend inwards.
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Concave quadrilateral
A concave quadrilateral is a four-sided polygon with at least one interior angle greater than 180 degrees, creating a dent in its shape.
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Square
A square is a quadrilateral with all four sides equal in length and all four angles equal to 90 degrees.
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Rectangle
A rectangle is a quadrilateral with four right angles, where opposite sides are equal in length.
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Parallelogram
A parallelogram is a quadrilateral with two pairs of parallel sides, where opposite sides are equal in length and opposite angles are equal.
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Rhombus
A rhombus is a quadrilateral with all four sides equal in length, and its diagonals bisect each other at right angles.
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Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel sides, often called the bases.
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Isosceles trapezoid
An isosceles trapezoid is a trapezoid with a pair of parallel sides and the non-parallel sides equal in length, making the base angles equal.
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Kite
A kite is a quadrilateral with two pairs of adjacent sides that are equal in length, and one pair of opposite angles equal.
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Opposite sides of a parallelogram
In a parallelogram, opposite sides are equal in length and parallel to each other.
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Opposite angles of a parallelogram
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary, adding up to 180 degrees.
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Diagonals of a parallelogram
The diagonals of a parallelogram bisect each other, dividing each diagonal into two equal parts.
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Area of a rectangle
The area of a rectangle is calculated by multiplying its length by its width.
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Area of a square
The area of a square is calculated by squaring the length of one of its sides.
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Area of a parallelogram
The area of a parallelogram is found by multiplying its base by its height.
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Area of a rhombus
The area of a rhombus is calculated by multiplying the lengths of its diagonals and dividing by two.
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Area of a trapezoid
The area of a trapezoid is calculated by multiplying the average of its two parallel sides by its height.
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Perimeter of a rectangle
The perimeter of a rectangle is the sum of all four sides, calculated as twice the sum of its length and width.
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Perimeter of a square
The perimeter of a square is four times the length of one of its sides.
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Sum of interior angles of a quadrilateral
The sum of the interior angles of any quadrilateral is always 360 degrees.
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Exterior angle of a quadrilateral
An exterior angle of a quadrilateral is formed by extending one side, and it equals the sum of the two non-adjacent interior angles.
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Diagonal of a rectangle
A diagonal of a rectangle connects two opposite corners and can be found using the Pythagorean theorem as the square root of the sum of the squares of the length and width.
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Diagonal of a square
A diagonal of a square connects two opposite corners and is calculated as the side length multiplied by the square root of 2.
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Properties of a kite
A kite has two pairs of adjacent equal sides, one pair of opposite equal angles, and its diagonals are perpendicular, with one diagonal bisecting the other.
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Cyclic quadrilateral
A cyclic quadrilateral is one that can be inscribed in a circle, meaning all its vertices lie on a single circle.
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Inscribed angle in a cyclic quadrilateral
In a cyclic quadrilateral, an inscribed angle is half the measure of the arc it subtends on the circle.
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Midline of a trapezoid
The midline of a trapezoid is a line segment connecting the midpoints of the non-parallel sides, and its length is the average of the lengths of the two parallel sides.
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Height of a trapezoid
The height of a trapezoid is the perpendicular distance between its two parallel sides.
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Common mistake with trapezoids
A common error is assuming a trapezoid has two pairs of parallel sides, but it actually has only one pair.
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Distinguishing rectangle and rhombus
A rectangle has all angles at 90 degrees but not necessarily equal sides, while a rhombus has all sides equal but not necessarily 90-degree angles.
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Pythagorean theorem in a rhombus
In a rhombus, the Pythagorean theorem applies to the right triangles formed by its diagonals, relating the side lengths and half-diagonals.
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Coordinate geometry for quadrilaterals
In coordinate geometry, quadrilaterals are plotted using points on a plane, and properties like slopes can determine parallelism or equal sides.
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Slope for parallel sides
In a quadrilateral, sides are parallel if they have the same slope when plotted on a coordinate plane.
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Distance formula for sides
The distance formula calculates the length of a side in a quadrilateral by finding the distance between two points on a coordinate plane.
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Varignon's theorem
Varignon's theorem states that connecting the midpoints of the sides of any quadrilateral forms a parallelogram.
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Example of square area
For a square with side length 4, the area is 16 square units.
If the side is 4, then area = 4 × 4 = 16.
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Example of rectangle diagonal
For a rectangle with length 6 and width 8, the diagonal is the square root of (6 squared plus 8 squared).
Diagonal = √(36 + 64) = √100 = 10.
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Example of parallelogram area
For a parallelogram with base 5 and height 3, the area is 15 square units.
Area = base × height = 5 × 3 = 15.
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Example of trapezoid area
For a trapezoid with parallel sides 4 and 6 and height 5, the area is the average of the bases times the height.
Area = [(4 + 6)/2] × 5 = 5 × 5 = 25.
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Rhombus diagonal intersection
In a rhombus, the diagonals intersect at right angles and bisect each other, dividing the rhombus into four right-angled triangles.
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Trapezoid leg properties
In an isosceles trapezoid, the legs are equal, and the base angles are equal, which helps in symmetry problems.
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Quadrilateral angle sum proof
The sum of interior angles in a quadrilateral can be proven by dividing it into two triangles, each with 180 degrees, totaling 360 degrees.
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Kite diagonal properties
In a kite, one diagonal is a line of symmetry and bisects the other diagonal at right angles.
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Perimeter of a rhombus
The perimeter of a rhombus is four times the length of one side, since all sides are equal.
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Cyclic quadrilateral opposite angles
In a cyclic quadrilateral, opposite angles sum to 180 degrees.
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Midpoint connector in quadrilateral
Connecting the midpoints of a quadrilateral's sides forms a parallelogram, as per Varignon's theorem.
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Example of kite area
For a kite with diagonals 10 and 4, the area is half the product of the diagonals.
Area = (10 × 4) / 2 = 20 square units.
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Parallel sides in coordinate plane
To verify parallel sides in a quadrilateral on a coordinate plane, check if the slopes of the lines are equal.
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Trapezoid with coordinates
A trapezoid can be defined by points on a plane where exactly one pair of sides has the same slope.
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Diagonals of a kite
The diagonals of a kite are perpendicular, and one bisects the other.
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Area comparison in quadrilaterals
Among quadrilaterals with the same perimeter, the square has the maximum area.
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Common trap with angles
A common error is assuming all angles in a quadrilateral are equal, which only applies to squares or rectangles.
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Height calculation in parallelogram
The height of a parallelogram is the perpendicular distance between its bases, not the length of its sides.
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Example of rhombus perimeter
For a rhombus with side length 7, the perimeter is 28 units.
Perimeter = 4 × 7 = 28.
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Quadrilateral inequality theorem
In any quadrilateral, the sum of the lengths of any three sides must be greater than the length of the fourth side.
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Isosceles trapezoid diagonals
In an isosceles trapezoid, the diagonals are equal in length.
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Example of cyclic quadrilateral
A rectangle is a cyclic quadrilateral because its vertices lie on a circle.
All angles are 90 degrees, summing opposites to 180.