Analytic Geometry — Quadrilaterals, Distance, and Triangle Types
Analytic Geometry — Quadrilaterals, Distance, and Triangle Types. Practice analytic geometry: identify parallelograms, rectangles, rhombi and trapezoids by coordinates, calculate distances, and classify triangles.
What is the area of the parallelogram?
Area of a parallelogram: \(S = b \cdot h = 10 \cdot 6 = 60\) square units
A parallelogram with base 8 cm and height 5 cm. What is its area?
Area: \(S = 8 \cdot 5 = 40\) square centimeters
Which of the following properties is true for a parallelogram?
In a parallelogram: opposite sides are equal and parallel, opposite angles are equal, and the diagonals bisect each other.
A parallelogram with area 72 cm² and height 9 cm. What is the base length?
From the formula: \(72 = b \cdot 9\), therefore \(b = 8\) cm
A parallelogram with vertices A(1,1), B(5,1), C(7,4), D(3,4). What is its area?
Base AB: \(|5-1| = 4\)
Height, the difference in y-values: \(|4-1| = 3\)
Area: \(S = 4 \cdot 3 = 12\) square units
What is the area of the rectangle?
Area of a rectangle: \(S = \text{length} \times \text{width} = 10 \cdot 5 = 50\) square units
A rectangle with length 12 cm and width 7 cm. What is its perimeter?
Perimeter of a rectangle: \(P = 2(a + b) = 2(12 + 7) = 38\) cm
A rectangle with area 48 cm² and length 8 cm. What is the width?
From the formula: \(48 = 8 \cdot b\), therefore \(b = 6\) cm
A rectangle with vertices A(1,2), B(6,2), C(6,5), D(1,5). What is its area?
Length: \(|6-1| = 5\), width: \(|5-2| = 3\)
Area: \(S = 5 \cdot 3 = 15\) square units
What is the length of the diagonal of a rectangle with sides 9 cm and 12 cm?
By Pythagoras: \(d = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = 15\) cm
Egyptian triangle: 9-12-15
What is the area of the square?
Area of a square: \(S = a^2 = 8^2 = 64\) square units
A square with area 49 cm². What is the side length?
From the formula: \(a^2 = 49\), therefore \(a = 7\) cm
A square with side 10 cm. What is its perimeter?
Perimeter of a square: \(P = 4a = 4 \cdot 10 = 40\) cm
A square with perimeter 36 cm. What is its area?
From the perimeter: \(4a = 36\), therefore \(a = 9\)
Area: \(S = 9^2 = 81\) square centimeters
What is the area of the rhombus?
Area of a rhombus: \(S = \frac{d_1 \cdot d_2}{2} = \frac{12 \cdot 16}{2} = 96\) square units
A rhombus with diagonals 10 cm and 8 cm. What is its area?
Area: \(S = \frac{10 \cdot 8}{2} = 40\) square centimeters
Which property is true for a rhombus?
In a rhombus: all 4 sides are equal, the diagonals are perpendicular and bisect each other, but they are not necessarily equal.
A rhombus with area 60 cm² and one diagonal 15 cm. What is the length of the other diagonal?
From the formula: \(60 = \frac{15 \cdot d_2}{2}\)
\(120 = 15 \cdot d_2\), therefore \(d_2 = 8\) cm
A rhombus with side 5 cm and one diagonal 8 cm. What is the length of the other diagonal?
What is the area of the kite?
Area of a kite: \(S = \frac{d_1 \cdot d_2}{2} = \frac{14 \cdot 10}{2} = 70\) square units
The formula is the same as for a rhombus.
A kite with diagonals 12 cm and 9 cm. What is its area?
Area: \(S = \frac{12 \cdot 9}{2} = 54\) square centimeters
What is the main difference between a rhombus and a kite?
In a rhombus, all 4 sides are equal. In a kite, there are two pairs of equal adjacent sides, but not necessarily all sides are equal.
Which property is shared by both a rhombus and a kite?
In both a rhombus and a kite, the diagonals are perpendicular to each other and form a 90-degree angle.
A kite with area 48 cm² and one diagonal 8 cm. What is the length of the other diagonal?
From the formula: \(48 = \frac{8 \cdot d_2}{2}\)
\(96 = 8 \cdot d_2\), therefore \(d_2 = 12\) cm
What is the area of the trapezoid?
Area of a trapezoid: \(S = \frac{(b_1 + b_2) \cdot h}{2} = \frac{(6 + 10) \cdot 8}{2} = 64\) square units
A trapezoid with bases 7 cm and 11 cm and height 5 cm. What is its area?
Area: \(S = \frac{(7 + 11) \cdot 5}{2} = \frac{90}{2} = 45\) square centimeters
An isosceles trapezoid is:
An isosceles trapezoid is a trapezoid whose legs, the non-parallel sides, are equal in length.
A trapezoid with area 60 cm², height 6 cm and one base 8 cm. What is the length of the other base?
From the formula: \(60 = \frac{(8 + b_2) \cdot 6}{2}\)
\(120 = (8 + b_2) \cdot 6\)
\(20 = 8 + b_2\), therefore \(b_2 = 12\) cm
The midsegment of a trapezoid (connecting the midpoints of the legs) is:
The midsegment of a trapezoid is parallel to the bases, and its length is \(m = \frac{b_1 + b_2}{2}\), the average of the bases.