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Geometric Mean (Altitude) Theorem

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the length of the two segments.

45°-45°-90° Triangle Theorem

In a 45°-45°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √2 times as long as each leg.

30°-30°-90° Triangle Theorem

In a 30°-30°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a convex n=gon is (n-2)·180°.

Corollary

The acute angles of a right triangle are complementary.

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex is 360°.

Rhombus Corollary

A quadrilateral is a rhombus if and only if it has four congruent sides.

Rectangle Corollary

A quadrilateral is a rectangle is and only if it has four right angles.

Square Corollary

A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Midsegment Theorem for Trapezoids

The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

Translation Theorem

A translation is an isometry.

Reflection Theorem

A reflection is an isometry.

Rotation Theorem

A rotation is an isometry.

Composition Theorem

The composition of two (or more) isometries is an isometry.

Reflection in Parallel Lines

If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation. I P" is the image of P, then:

Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is one half the measure of its intercepted arc.

Angles Inside the Circle

If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arc intercepted by the angle and its vertical angle.

Angles Outside the Circle

If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measure of the intercepted arcs.

Segments of Chords Theorem

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Segments of Secants and Tangents Theorem

If a secant segment and a tangent segment share and endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Area of a Rectangle

The area of a rectangle is the product of its base and height. A=bh

Area of a Parallelogram

The area of a parallelogram is the product of a base and its corresponding height. A=bh

Area of a triangle

A = ½ bh (or) A = bh÷2

Area of a trapezoid

1/2 (b+b)h or mh

Area of a Rhombus

The area of a rhombus is one half the product of the lengths of its diagonals. A=1/2d1d2

Area of a Kite

The area of a kite is one half the product of the lengths of its diagonals. A=1/2d1d2

Areas of Similar Polygons

If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a ²:b ².

Circumference of a Circle

π times the diameter

Arc Length Corollary

In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.

Area of a Circle

pi r ^ 2

area of a sector

a fractional part of the area of a circle

Area of a Regular Polygon

The area of a regular n-gon with sides length s is half the product of the apothem a and the peroimeter P, so A=1/2aP, or A=1/2a*ns.

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.

Surface Area of a Right Prism

The surface area S of a right prism is the sum of the base areas and the lateral area, S=2B=Ph, where B is the area of a base, P is the perimeter of a b…

Surface Area of a Regular Pyramid

The surface area S of a regular pyramid is S=B=1/2Pl, where B is the area of the base, P is the perimeter of the base, and l is eh slant height.

surface area of a right cone

The surface area S of a right cone is S=B=1/2Cl=╥r^2=╥rl, where B is the area of the base, C is the circumference of the base, r is the radius of the base, and l is the slant height.

Volume of a Prism

The volume V of a prism is V=Bh, where B is the area of a base and h is the height.

Volume of a Cylinder

V = pi r squared h

Cavalieri's Principle

Same Base - Same Height - Volumes are the Same.

Volume of a Pyramid

V = 1/3 lwh

Volume of a Cone

V = 1/3 pi r squared h

Surface Area of a Sphere

The surface area S of a sphere with radius r is S=4/3╥r^3.

volume of a sphere

V = 2/3 pi r squared h (or) V = 4/3 pi r cubed

similar solids theorem

If two similar solids have a scale factor of a:b, then corresponding areas have a ratio of a ²:b ², and corresponding volumes have a ratio of a³:b³.