Calculator Techniques for Quadrilaterals in Plane Geometry
What Is a Quadrilateral?
A quadrilateral also called as "Tetragon" or "Quadrangle", is a general term for any polygon with four sides. There are six types of quadrilaterals. They are square, rectangle, parallelogram, trapezoid, rhombus, ad trapezium.
Common Parts of a Quadrilateral
The common parts of a quadrilateral are described as follows.
1. Side - A side is a line segment which joins any two adjacent vertices.
2. Interior Angle - An interior angle is the angle formed between two adjacent sides.
3. Height or Altitude - It is the distance between two parallel sides of a quadrilateral
4. Base - This is the side that is parallel to the altitude.
5. Diagonal - This is the line segment joining any two non-adjacent vertices
A special type of a rectangle in which all the sides are equal.
A = a^2
P = 4a
A parallelogram in which the interior angles are all right angles.
A = ab
P = 2b + 2a
A parallelogram in which all sides are equal.
A = (1/2) (d1) (d2)
P = 4a
A quadrilateral with one pair of parallel sides.
A = (1/2) (a + b) (h)
P = a + b + c + d
A quadrilateral with no parallel sides.
P = a + b + c + d
A quadrilateral in which the opposite sides are parallel.
A = bh OR A = absin(theta)
P= 2a + 2b
Properties of a Parallelogram
1. The opposite sides of a parallelogram are equal, and so also the opposite angles.
2. The diagonals of a parallelogram bisect each other.
3. If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is therefore a parallelogram.
A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle. According to Ptolemy's theorem, the product of the diagonals of any cyclic quadrilateral is equal to the sum of the products of the opposite sides.
d1d2 = (a) (c) + (b) (d)
Problem 1: Area of a Rhombus
A rhombus has diagonals of 32 and 20 inches. What is the area?
a. Given the length of the rhombus' diagonals, use the formula in solving for the area.
d1 = 32 inches d2 = 20 inches Area of rhombus = 1/2 (d1) (d2) Area of rhombus = 1/2 (32) (20) Area of rhombus = 320 square inches
Final Answer: The area of the rhombus is 320 square inches.
Problem 2: Square Inscribed in a Semicircle
Find the difference of the area of the square inscribed in a semicircle having a radius of 15 centimeters.
a. Calculate the side length of the square inside the semicircle.
a^2 + b^2 = c^2 x^2 + (x/2)^2 = 15^2 x^2 + (x^2/4) = 225 5/4 x^2 = 225 x = 180 centimeters x = 13.42 centimeters
b. Solve for the area of the semicircle.
Area of semicircle = (pi) (r)^2 / 2 Area of semicircle = (pi) (15)^2 / 2 Area of semicircle = 353.49 square centimeters
c. Solve for the area of the square inscribed.
Area of square = x^2 Area of square = (13.42)^2 Area of square = 180.10 square centimeters
d. Get the difference of areas.
Area difference = Area of semicircle - Area of square Area difference = 353.49 - 180.10 Area difference = 173.39 square centimeters
Final Answer: The difference of the area of the square inscribed in a semicircle is 173.39 square centimeters.
Problem 3: Lengths of Trapezoid's Bases
A trapezoid has an area of 36 square centimeters and a height of 2 centimeters. Its two bases have a ratio of 4: 5. What are the lengths?
a. Solve for the area of the trapezoid.
Altitude of trapezoid (h) = 2 centimeters Area of trapezoid = 36 square centimeters Area of trapezoid = (1/2) (b1 + b2) (h) 36 square centimeters = (1/2) (b1 + b2) (2)
b. Let b1 = 4x and b2 = 5x. Solve for the value of x.
36 square centimeters = (1/2) (b1 + b2) (2) 36 square centimeters = (1/2) (4x + 5x) (2) 36 square centimeters = 4x + 5x 36 square centimeters = 9x x = 4 centimeters
c. Solve for b1 and b2 by substituting the value of x.
4x = 4 (4) 4x = 16 centimeters 5x = 5 (4) 5x = 20 centimeters
Final Answer: The lengths of the trapezoid's bases are 16 and 20 centimeters.
Problem 4: Diagonals and Angles of a Rhombus
One of the diagonals of the rhombus is 12 inches. If the area of the rhombus is 132 square inches, determine the following.
a. length of the other diagonal
b. the measure of acute angles in degrees between the two sides
a. Solve for the length of other diagonal using the formula for area.
Area of rhombus = 1/2 (d1) (d2) 132 square inches = 1/2 (12 inches) (d2) d2 = 22 inches
b. Solve for the length of one side of the rhombus using the Pythagorean theorem.
X^2 = 11^2 + 6^2 X = 12.53 inches
c. Solve for the angle θ.
tan θ = opposite / adjacent tan θ = 6/11 θ = 28.61 degrees
Final Answer: The length of the other diagonal is 22 inches, and the angle between the two sides is 28.61 degrees.
Problem 5: Interior Angles of Parallelogram
Two sides of parallelogram measure 68 centimeters and 83 centimeters and its shorter diagonal is 42 centimeters. Calculate the hugest interior angle.
a. Using the cosine law, solve for the θ between the two known sides.
42^2 = 68^2 + 83^2 – 2(68)(83) cosine (θ) θ = 30.27 degrees
b. Solve for the supplementary angle of θ.
Largest interior angle = 180 – 30.27 degrees Largest interior angle = 149.73 degrees
Final Answer: The interior angle of the rhombus with the highest value is 149.73 degrees.
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