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Math Study Guide Semester 2 Final KAMERON (Unit 6 (Ratio: The relationship…

- - - - Cross-Product property: the product of the extremes = the product of it means. If a/b = c/d, then ab = bc.
        
        Reciprocal property: If the ratios are = then their reciprocals are =
        If a/b = c/d, then b/a = d/c
        
        Geometric Mean: An average where you multiply 2 numbers and then square root them. Formula: a/x = x/b, or x² = ab, or x = √ab
        
        Scale: ratio that describes the dimensions in the drawing to the actual dimensions of the object
        
        Polygon: A plane figure formed by 3 or more segments calles sides.
        
        Similar Polygons: 2 polygons are similar if corresponding angles are congruent and corresponding side lengths are proportional.
        
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- - - - Convex: When there is no line on a polygon that has a point on the inside of the polygon.
        Concave: A polygon that is not convex.
        Draw
        
        Equilateral: When all sides are congruent Equiangular: When all angles are congruent Regular:When a Polygon is both equilateral and equiangular Draw
        
        Diagonal of a Polygon: A segment that joins two nonconsecutive vertices. Draw
        
        Sum of Interior Angles: The number of sides of the polygon subtracted by 2 then multiplied by 180° Ex: Pentagon = 5 sides - 2= 3 x 180° = 540° Draw
        
        Polygon Interior Angles Theorem: The sum of the interior angles of a convex n-gon is (n - 2) x 180° Draw
        
        Corollary:The measure of each interior angle of a regular n-gon is: Formula: (n-2) x 180° / n, or 360° / n Draw
        
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- - - - Arc Length Corollary: In a circle the ratio of a given arc to the circumference is equal to the ratio of the measure of the arc to 360° Draw
        
        Perimeter: The distance around a 2 dimensional shape
        
        Postulates -
        Area of a Square Postulate: The area of a square is the square of the length of its side, or A = S²
        Area Congruence Postulate: If2 polygons are congruent then they have the same area.
        Area Addition Postulate: The area of a region is the sum of the areas of its overlapping parts.
        Draw
        
        Sector of a Circle: The region bounded by two radii of the circle and their intercepted arc. (Looks like a slice of pice)
        Area of a Sector: The ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360°
        Formula: Area of Sector/ πr² = m of arc/ 360°
        Draw
        
        Center: Center of the circumscribed circle
        Radius: Radius of the circumscribed circle
        Apothem: The distance from the center to any side of the polygon
        Draw
- - - - Converse of Pythagorean Theorem: If the hypotenuse of a triangle is = to the sum of both legs, then it is a right triangle.
        
        Classifying Angle Theorems:
        Acute: If the hypotenuse is less then the sum of the legs, then it is an acute triangle. Formula: c² < a² + b² then it is acute.
        Obtuse: If the hypotenuse is greater then the legs, then it is an obtuse triangle. Formula: c² > a² + b² then it is obtuse.
        Draw
        
        Altitude to Hypotenuse Similarity Theorem: If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle and to each other. (Father, Mother and Baby) Draw
        
        Geometric Mean: An average where you multiply 2 numbers and then square root them. Formula: a/x = x/b, or x² = ab, or x = √ab
        
        Geometric Mean Theorems for Right Triangles:
        The Roof Top Formula: h= L1 x L2 /H
        Altitude: BD/CD = CD/AD
        Right Leg: AB/CB = AC/AD
        Left Leg: AB/AC = AC/AD
        Draw Triangle:
        
        45° - 45° - 90° Triangle Theorem: In a triangle with 2 angles of 45° and one of 90°, the hypotenuse is √2 times as long as each leg. Draw
        
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- - - - Common Tangent: The line or segment that is a tangent to two coplanar circles.
        Common internal Tangent: It intersects the segment that joins the center of the two circles.
        Common external Tangent: It does not intersect the segment that joins the centers of the two circles.
        Draw
        
        Tangent Thorems -
        Theorem 10.1: In a plane, a line is a tangent to a circle if and only if the line is perpendicular to a radius of a circle at its endpoint on the circle.
        Theorem 10.2: Tangent segments that form a common external point are congruent.
        Chord Theorems -
        Theorem 10.3: In the same circe or in congruent circles, 2 minor arcs are congruent if and only if their corresponding chords are congruent.
        Theorem 10.4: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
        Theorem 10.5:If a diameter of a circle is perpendicular to a chord then the diameter bisects the chord and it arc.
        Theorem 10.6: In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from its center.
        Theorem 10.7:The measure of a inscribed angle is half the measure of its intercepted arc is an angle whose vertex is on a circle and whose sides contain chords.
        Theorem 10.8:If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent.
        Theorem 10.9: If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of a circle. Conversely, if one side of and inscribed triangle is a diameter of the circle then the triangle is a right triangle and the angle opposite to the diameter is a right angle.
        Theorem 10.10: A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
        Postulates -
        Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
        Congruent Arcs: Two arcs are congruent if they have the same measure and they are acrs of the same circle or of congruent circles.
        Draw
        
        Arc Addition Postulate:
- - - - Pentagonal Prism: The two bases of a prism are congruent polygons in parallel planes.
        Triangular Pyramid: The base of a pyramid is a polygon.
        
        Regular Polyhedron: all of its faces are congruent regular polygons.
        Convex Polyhedron: Any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.
        Draw
        
        Euler's Theorem: The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula: F + V = E + 2
        
        Volume: The number of cubic units contained in a solid's interior
        
        Postulates:
        Volume of a cube Postulate: The volume of a cube is the cube of the length if its side V=S3 (cubed)
        Volume Congruence Postulate: If two polyhedra are congruent, then they have the same volume.
        Volume Addition Postulate: The volume of a solid is the sum of the volumes of all its non-overlapping parts.
        
        Prism: A polyhedron with two congruent faces, called bases, that lie in parallel planes. The other faces, called lateral faces are parallelograms formed by connecting the corresponding vertices of the bases. The segments connecting these vertices are lateral edges.
        Volume of a Prism: V = Bh
        Right Prism: Each lateral edge is perpendicular to both bases.
        Oblique Prism: Lateral edges are not perpendicular to both bases. The length of the oblique lateral edges is the slant height of the prism.
        Surface Area of Right Prism:
        S = 2B + Ph
        Draw
        
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