Plane and Solid Geometry Midterm Reviewer PDF

Plane and Solid Geometry Midterm Reviewer 2SEDM-A Week 7: Polygons Area - Space enclosed by the polygons ‘Polygon’ is a Greek word. ‘Poly’ means many and ‘gon’ means angles. - The...

Plane and Solid Geometry Midterm Reviewer 2SEDM-A Week 7: Polygons Area - Space enclosed by the polygons ‘Polygon’ is a Greek word. ‘Poly’ means many and ‘gon’ means angles. - The word polygon means “many angles” - A two-dimensional object - A closed figure - Made up of three or more straight line segments - There are exactly two sides that meet at a vertex - The sides do not cross each other Triangle – 3 sides (e.g., equilateral, isosceles, scalene) Quadrilateral – 4 sides (e.g., square, rectangle, trapezoid) Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Side: One of the line segments Decagon – 10 sides that make up a polygon. Vertex: Point where two sides meet. Week 8: Triangles and Congruent Triangle Interior angle: formed by two adjacent sides inside the polygon. Classification of Triangles: Classification of Triangles by Sides: Formula for the total sum of Interior Angles in any - Scalene: No congruent sides polygon: - Isosceles: Two congruent sides (𝑛 − 2)180° - Equilateral: All sides are congruent Formula for the measurement of each interior angles in a Classification of Triangles by Angles: regular polygon: - Acute Triangle: All interior angles are acute (less than 90 ((𝑛 − 2)180°) degrees) - Right Triangle: One interior angle is Right Angle (Exactly 𝑛 90 degrees) Exterior angle: formed by two adjacent sides outside the - Equiangular Triangle: All interior Angles are congruent polygon. (The sum of exterior angles is always equal to 360 - Obtuse Triangle: One interior angle is Obtuse (More than degrees) 90Degrees, but less than 180 degrees) Diagonal - a line segment connecting two non-adjacent vertices. Properties of Triangles: Sum of Interior Angles: The sum Types of Polygon of interior angles is always equal Equiangular Polygon: all of the angles are equal to 180 degrees. Equilateral Polygon: all of the sides are same length Exterior Angle Theorem: Exterior Regular Polygon: all the angles are equal, and all of Angles is equal to the sum of two the sides are the same length. They are both Opposite interior angles. equilateral and equiangular Triangle Inequality Theorem: The sum of the lengths of any two Convex - All interior angles are less than 180° sides of a triangle must be greater’ Concave - At least one interior angle is greater than 180° than the length of the third side. Simple - Sides do not intersect or cross each other. Complex - Self intersecting (Ex. Star) Regular - All sides and angles are equal Irregular - Has sides and angles of different length and degrees. Perimeter - The total length of the sides K.A.D. Plane and Solid Geometry Midterm Reviewer 2SEDM-A Week 9: Quadrilateral Congruent Triangles: SSS (Side-Side-Side): If three sides of one triangle are Formula for the area of different Quadrilaterals: congruent to three sides of another triangle, then the Square: two triangles are congruent. 𝐴 = 𝑠2 Rectangle: 𝐴= 𝑙 ×ℎ Rhombus: 1 𝐴 = (𝑑1 )(𝑑2 ) 2 SAS (Side-Angle-Side): If two sides and the included Kite: angles of one triangle are congruent to two sides and 1 the included angle of another triangle, then the 𝐴 = (𝑑1 )(𝑑2 ) 2 triangles are congruent. Trapezoid: 1 𝐴 = (𝑑1 )(𝑑2 ) ℎ 2 Parallelogram: 𝐴= 𝑏 ×ℎ ASA (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two Diagonal Theorem for Parallelograms: The diagonals of a angles and the included side of another triangle, then parallelogram bisect each other. the triangles are congruent. Diagonal Theorem for Rectangle: The diagonals of a AAS (Angle-Angle-Side): If two angles and a non- rectangle are equal in length and bisect each other. included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. Diagonal Theorem for Rhombus: The diagonals of a rhombus bisect each other at right angles (90 degrees). HL (Hypotenuse-Leg): In a Right Triangle, if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent. K.A.D. Plane and Solid Geometry Midterm Reviewer 2SEDM-A Diagonal Theorem for Square: The diagonals of a square Week 10: Circles are equal in length, bisect each other, and intersect at right angles. A Circle is defined as a set of points that are all the same distance away from a central point, called the center. Radius of a Circle: The distance r from the center to any point on the circle. Symbols of a Circle: We use the symbol ‘O’ to represent the center of the circle, ‘r’ to represent the radius of the circle, and ‘ ‘ represent the arc of the circle. Diameter Diameter of a circle: A line segment that passes through the center of the circle Midsegment Theorem for Trapezoid: The Midsegment and has endpoints on the circle. The Theorem for trapezoids states that the segment diameter is twice as long as the radius. connecting the midpoints of the non-parallel sides (the Arcs of Circle: A portion of the circle between two legs) of a trapezoid is called the midsegment. points on the circle. Arc The midsegment is parallel to the bases of the - Minor Arc: An arc that is less trapezoid. than 180 degrees. The length of the midsegment is equal to the average - Semi-Circle: An arc that is of the lengths of the two bases. exactly 180 degrees. - Major Arc: An arc that is more Chord than 180 degrees, but less than 360 degrees. Secant Chord: A line segment that connects two points on the circle’s edge. Secant: A line that intersects the circle at two points. Sector: A region of the circle that is bounded by two radii and an arc. Minor Sector - Major Sector: Large Area of a Formula for the length of the Midsegment: Circle Major - Minor Sector: Small Area of a 1 Circle 𝑚= (𝑏 + 𝑏2 ) 2 1 Segment: A region of the circle that Segment is bounded by a chord and an arc. Theorem for Kites: Tangent: A line that touches the circle Diagonal Theorem: In a kite, the diagonals are at exactly one point, called the point perpendicular to each other. The diagonal that of tangency. Point of Tangent connects the vertices of the unequal angles bisects Intercepted Arc: An arc that Tangency the other diagonal. lies between two intersecting lines Angle Theorem: The angles between the unequal that intersect inside the circle Intercepted sides of a kite are equal. Arc Side Theorem: In a kite, the two pairs of adjacent sides are equal. A sphere is the set of all points in 3D space that are Area Theorem: The area of a kite can be calculated equidistant from a fixed point (the center). using the lengths of its diagonals. Radius of a Sphere: The distance from the center to any point on the sphere. Diameter of a Sphere: Twice the radius, passing through the center and connecting two points on the sphere. Great Circle Great Circle: A circle on a sphere that has the same diameter as the sphere itself, dividing the sphere into two equal hemispheres. K.A.D. Plane and Solid Geometry Midterm Reviewer 2SEDM-A Tangent-Secant Theorem: If a tangent and a secant are drawn from a common external point to a circle, then the square of the length of the tangent segment is equal to the product of the entire length of the secant segment and the length of its external part. Angle of Intersection Theorem: When two chords intersect inside a circle, the measure of the angle formed by the intersection of the chords is equal to half the sum of the measures of the arcs intercepted by the angle. Formula: 𝑡 2 = 𝑆 × 𝑒 Central Angle Theorem: The measure of a central angle in a circle is equal to the measure of its intercepted arc. Formula for the Area of a Circle: 𝐴 = 𝜋 𝑟2 Where: A is the area of the circle π (pi) is approximately 3.141593.141593.14159 r is the radius of the circle (the distance from the center of the circle to any point Inscribed Angle Theorem: The measure of an inscribed on its circumference). angle in a circle is half the measure of its intercepted arc. Area of a Sector of a Circle: θ A= π r2 360° Where: A is the area of the Sector 𝜽 is the central angle of the sector in degrees π (pi) is approximately 3.141593.141593.14159 r is the radius of the circle (the distance from the center of the circle to any point on its circumference). Tangent-Chord Theorem: The angle formed between a tangent and a chord drawn from the point of tangency is Area of a triangle inscribed in a circle: equal to half the measure of the intercepted arc created abc by that chord. A= 4R K.A.D. Plane and Solid Geometry Midterm Reviewer 2SEDM-A Where: Area (2D Shapes): A is the area of the Sector Triangle: a,b, and c are the lengths of the triangle's 1 𝐴= 𝑏ℎ sides 2 R is the circumradius (the radius of the Square: circumcircle that passes through all three 𝐴 = 𝑠2 vertices of the triangle). Rectangle: Formula for the surface area of a sphere: 𝐴 =𝑙×w A = 4 π r2 Rhombus: 1 𝐴= × 𝑑1 𝑑2 Where: 2 A is the surface area of the sphere Parallelogram: r is the radius of the sphere 𝐴=𝑏×ℎ π (pi) is approximately Kite: 3.141593.141593.14159 1 𝐴 = × 𝑑1 𝑑2 Area of circular segment formed by a chord and the arc 2 of the circle: formed by a chord and the corresponding arc Trapezoid: of a circle can be calculated using the following formula: 1 𝐴= (𝑎 + 𝑏) × ℎ 2 r2 Circle: A = (θ − sinθ) 2 𝐴 = 𝜋 𝑟2 Where: A is the area of the circular segment, Definition of Basic Solids: r is the radius of the circle, Characteristics of a Cube: A cube is a three- Θ or theta is the central angle in radians dimensional solid with six equal square faces, twelve corresponding to the arc. equal edges, and eight vertices. All its angles are right angles. - Faces: 6 square face Week 11: Perimeter, Area, and Solids and Basic - Edges: 12 edges of equal length Geometric Constructions - Vertices: 8 vertices where 3 edges meet Volume Formula: Perimeter (2D Shapes): Triangle: 𝑉 = 𝑠3 𝑃 =𝑎+𝑏+𝑐 Surface Area Formula: Square: 𝐴 = 6 𝑠2 𝑃 =4s Prism: Rectangle: - Rectangular Prism: A three-dimensional solid 𝑃 =2 (l + w) with six rectangular faces, opposite faces being equal. All angles are right angles. Rhombus: - Triangular Prism: A prism with two parallel, 𝑃 =4s congruent triangular bases and three rectangular faces Parallelogram: connecting the bases. - Pentagonal Prism: A prism with two parallel, 𝑃 = 2 (l + b) congruent pentagonal bases and five rectangular faces Kite: connecting the bases. 𝑃 = 2 (a+ b) - Hexagonal Prism: A prism with two parallel, Trapezoid: congruent hexagonal bases and six rectangular faces connecting the bases. 𝑃 =a+b+c+d - Other Prisms (n-sided): A prism with two parallel, congruent n-sided polygonal bases, connected by n- rectangular faces. K.A.D. Plane and Solid Geometry Midterm Reviewer 2SEDM-A Volume Formula for Prisms: Surface Area Formula: 𝑉 =𝑙×ℎ 𝐴 = 4 𝜋 𝑟2 Surface Area Formula: Pyramid: A pyramid is a solid with a polygonal base and triangular faces that meet at a single vertex (apex). The type of pyramid is named after the shape of its base (e.g., triangular pyramid, square pyramid). - Base: A polygonal base (triangular, square, pentagonal, etc.) - Faces: Triangular faces converging at the apex Volume Formula for Prisms: Cylinder: A cylinder is a three-dimensional solid with two parallel, circular bases connected by a curved surface. The axis of the cylinder is the line segment joining the centers of the two circular bases. - Bases: 2 circular bases of equal radius - Curved Surface: One continuous, curved surface connecting the bases Volume Formula for Prisms: 𝑉 = 𝜋 𝑟2ℎ Surface Area Formula: Surface Area Formula: 𝐴 = 2 𝜋 𝑟(ℎ + 𝑟) Cone: A cone is a three-dimensional solid with a circular base and a single vertex not in the plane of the base. The lateral surface slopes from the base to the apex. - Base: 1 circular base - Vertex: A single vertex (apex) where the curved surface converges Volume Formula for Prisms: 1 𝑉= 𝜋 𝑟2ℎ 3 Surface Area Formula: 𝐴 = 𝜋 𝑟(𝑟 + 𝑙) Sphere: A sphere is a perfectly round three- dimensional solid, where all points on its surface are equidistant from its center. - Surface: Continuous, curved surface with no edges or vertices Volume Formula for Prisms: 4 𝑉= 𝜋 𝑟3 3 K.A.D.