Theorems, Postulates, Corollaries, and Important Terms after Theorem 8-2

Vector

Any quantity such as force, velocity, or acceleration, that has both magnitude(size) and direction, is a vector.(Symbol is arrow on top of name of vector but only with half the arrow, the part pointing upward)

Base

Any side of a rectangle or other parallelogram can be considered to be a base.

Adjacennt arcs

Arcs that have exactly one point in common.

Congruent arcs

Arcs, in the same circle or in congruent circles, that have equal measures.

Circumferencee, C, of circle with radius r

C=2πr

Circumferencee, C, of circle with diameter d

C=πd

center of a regular polygon

Center of the circumscribed circle.

Congruent Circles/Spheres

Circles or spheres that have congruennt radii.

Concentric circles/spheres

Circles that lie in the same planne annd have the same center. Concentric spheres are spheres that have the same cennter.

Tangent Circles

Coplanar circles that are tangent to the same line at the same point.(blue is internally tangent red/pink is externally tangent)

Radius of a regular polygon

Distance from the center to a vertex.

Theorem 11-2(Area of parallelogram)

The area of a parallelogram equals the product of a base and the height to that base .(A=bh)

Theorem 11-1(Rectangle Area)

The area of a rectangle equals the product of its base and height (A=bh)

Postulate 19 Area Addition Postulate

The area of a region is the sum of the area of its non-overlapping parts.

Theorem 11-6(Area of a regular polygon)

The area of a regular polygon is equal to half the product of the apothem and the perimeter. (A=1/2ap)

Theorem 11-4(Area of Rhombus)

The area of a rhombus equals half the product of its diagonals.(A=1/2d1d2)

Theorem 12-9(Area of a Sphere)

The area of a sphere equals 4π times the square of the radius. (A=4πr^2)

Postulate 17 (Square)

The area of a square is the square of the length of a side. (A=s^2)

Theorem 11-5(Area of Trapezoid)

The area of a trapezoid equals half the product of the height and the sum of the bases. (A=1/2h(b1+b2))

Theorem 11-3(Area of triangle)

The area of a triangle equals half the product of a base and the height to that base. (A=1/2bh)

Bases of a Prism

The bases are congruent polygonns lying in parallel planes.

Theorem 10-1(Bisectors of the angles of a triangle meet where that are blank form the three sides?)

The bisectors of the angles of a triangle intersect in a point that is equidistant from the three sides of the triangle.

Circle circumscribed about the polygon

The circle is circumscribed about the polygon when each vertex of the polygon lies on the circle. (Circle contains polygon)

Theorem 14-6(What is a composite of two isometries)

The composite of two isometries is an isometry

Theorem 13-1 The Distance Formula

The distance d between points (x1, y1) and (x2, y2) is given by: d=√(x2-x1)^2+(y2-y1)^2

Lateral Faces of a prism

The faces of a prism that are not its bases are called lateral faces.

Theorem 13-6 Standard Form

The graph of any equation that can be written in the form Ax+By=C where A and B are not both zero, is a line.

Theorem 12-7(Lateral Area of a cone)

The lateral area of a conne equals half the circumference of the base times the slant height. (L.A.=1/2 *2πr*l, or L.A.=πrl)

Theorem 12-5(Lateral area of a cylinder)

The lateral area of a cylinder equals the circumference of the base times the height of the cylinder. (LA=2πrh)

Theorem 12-3(Formula for lateral areea of a regular pyramid)

The lateral area of a regular pyramid equals half the perimeter of the base times the slant height. (L.A. = 1/2 pl)

Theorem 12-1(Lateral Area of a Right Prism)

The lateral area of a right prism equals the perimeter of the base times the height of the prism (L.A. = Ph)

Lateral Area of a Prism

The lateral area(L.A.) of a prism is the sum of the areas of its lateral facese.

Height

The length of an altitude is called the height.

Height Of the Prism

The length of an altitude is the height, h, of the prism.

Theorem 10-3(The lines that contain the blank intersect in a point)

The lines that contain the altitudes of a triangle intersect in a point

Magnitude of a Vector

The magnitude of a vector AB is the length of the arrow from point A to point B is denoted by the symbol |AB|.

Theorem 9-8(Relationship between angle consisting of chord and tangent to the incepted arc)

The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc

Theorem 9-9(Relationship between angle formed by two chords that intersect inside a circle and the intercepted arcs)

The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs

Theorem 9-10(Relationship between measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle and the intercepted arcs)

The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs

Theorem 9-7 (Relationship of measure between inscribed angle and intercepted arc)

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

Measure of a Minor Arc

The measure of its central anngle.

Postulate 16 (Arc Addition Postulate)

The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.

Theorem 10-4(relationship between segment that is created when medians of a triangle intersect at a point and the median)

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

Theorem 13-5 The Midpoint Formula

The midpoint of the segment that joins points (x1,y1) and (x2,y2) is the point (x1+x2/2, y1+y2/2).

Inverse

The opposite of a mapping is the inverse. noted by e.g. T and T^-1

Theorem 10-2(Perpendicular bisectors of sides of a triangle meet where?)

The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the three vertices of the triangle

Preimage

The preimage is the original before the mapping

Altitude of a Pyramid

The segment from the vertex perpendicular to the base is the altitude and its length is the height, h, of the pyramid.

Slope

The slope, denoted by m, of the nonvertical line through the points (x1,y1) and (x2,y2) is defined as follows:slope m=(y2-y1)/(x2-x1)=change in y/change in x

Total Area Of a Prism

The total areea is the sum of the areas of all its faces.

Theorem 12-8(Volume of a Cone)

The volume of a cone equals one third the area of the base times the height of the cone (V =1/3πr^2h)

Theorem 12-6(Volume of a Cylinder)

The volume of a cylinder equals the area of a base times the height of the cylinder. (V=πr^2h)

Theorem 12-4(Volume of a pyramid)

The volume of a pyramid equals one third the area of the base times the height of the pyramid. (V = 1/3 Bh)

Theorem 12-2(Volume of a right prism)

The volume of a right prism equals the area of a base times the height of the prism (V=Bh)

Theorem 12-10(Volume of a Sphere)

The volume of a sphere equals 4/3π times the cube of the radius. (V=4/3πr^3)

Identity

Thee mapping that maps every point to itself is called the identity transformation I

Lateral Edges of a Pyramid

These faces intersect in segments called lateral edges.

Line Symmetry

This means that for each figure there is a symmetry line k such that the reflection Rk maps the figure onto itself.

Point Symmetry

This means that for each figure there is a symmetry poinnt O such that the half turn Ho maps the figure onto itself.

How to decide whether solids are similar

To decide whether two other solids are similar, determine whether bases are similar and corresponding lengths are proportional.

Method 1 of Finding the Lateral Area of a Regular Pyramid

To find the lateral area of a regular pyramid with n lateral faces: Find the area of one lateral face and multiply by n.

Method 2 of Finding the Lateral Area of a Regular Pyramid

To find the lateral area of a regular pyramid with n lateral faces: Use the formula L.A.=1/2pl.

Lateral Edges of a Prism

Adjacent lateral faces intersect in parallel segments called lateral edges.

Altitude of a Trapezoid

An altitude of a trapezoid is any segment perpendicular to a line containing a base from a point on the opposite base.

Altitude

An altitude to a base is any segment perpendicular to the line containing the base from any point onn the opposite side.

Central angle of a regular polygon

An angle formed by two radii drawn to consecutive vertices.

9-7 Corollary 2 (What type of angle is an angle that is inscribed in a semicircle)

An angle inscribed in a semicircle is a right angle

Inscribed Angle

An angle whose vertex is on a circle and whose sides contain chords of the circle. (When angles intercept arcs, you can think of intercept as create.)

Central Angle of a Circle

An angle with its vertex at the center of the circle.

Theorem 13-2(Equation of circle with center(a,b) and radius r)

An equation of the circle with center (a,b) and radius r is (x-a)^2 + (y-b)^2 = r^2

Corollary 2(What does isometry do to a polygon's area)

An isometry maps a polygon to a polygon with the same area.

Theorem 14-1(What does isometry do to triangle)

An isometry maps a triangle to a congruent triangle.

Corollary 1(What does isometry do to angle)

An isometry maps an angle to a congruent angle.

Theorem 13-4(What must product of slopes be if lines are perpendicular)

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. m1 * m2 = -1, or m1 = -1/m2

Theorem 13-3(What must slopes be if lines are parallel)

Two nonvertical lines are parallel if and only if their slopes are equal.

Equal Vectors

Two vectors are equal if they have the same magnitude and the same direction.

Parallel Vectors

Two vectors are parallel if the arrows representing them have the same direction or opposite directions. Also, two vectors can be parallel even if they're collinear.

Adding Vectors

Vectors can be added by the followinng simple rule:(a,b)+(c,d)=(a+c,b+d).

Theorem 9-13(Relationship between tangent segment and the secant segment and its external segment)

When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment

Theorem 9-11(Relationship between segments of the chords when multiplied)

When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.

Concurrent

When two or more lines intersect in one point, the lines are said to be concurrent.

Theorem 9-12(Relationship between two secant segments and external segments when multiplied)

When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.

Major Arc

YWZ

Minor Arc

YZ

Theorem 13-8 Point-Slope Form

an equation of the line that passes through the point (x1,y1) and has slope m is y-y1 = m(x-x1)

Cosine

cosine of angle=leg adjacent angle/hypotenuse

Incenter of a Triangle

point where angle bisectors meet

Circumcenter of a Triangle

point where perpendicular bisectors meet

Sine

sine of angle=leg opposite angle/hypotenuse

Tangent

tangent of angle a=leg opposite angle a/leg adjacent angle a.

Area of sector

x/360 πr²

Common Right Triangle Lengths

(3,4,5,)(6,8,10)(9,12,15)(12,16,20)(15,20,25)(5,12,13)(10,24,26)(8,15,17)(7,24,25)

Coordinates after a 90 degrees counterclockwise rotation

(a,b)-->(-b,a)

Coordinates of a Vector

(change in x, change in y)

Formula for Measure of Int. Angle

(n-2)180/n

Rotational Symmetry

1. 90 degrees rotational symmetry 2.180 degrees rotational symmetry/Ho 3. 270 degrees rotational symmetry 4. 360 degrees rotational symmetry/ the identity I Note 180 degrees rotational symmetry is another name for point symmetry.

Construction 12 (Given a segment, divide the segment into a given number of congruent parts.)

1. Choose any point Z not on AB(line). Draw AZ(ray). 2. Using any radius, start with A as cnter and mark off R, S, and T so that AR=RS=ST. 3. Draw TB(segment). 4. At R and S construct lines parallel to TB, intersecting AB(segment) in X and Y.

Construction 11(Given a triangle, inscribe a circle in the triangle.)

1. Construct the bisectors of angle A and angle B. Label the point of intersection I. 2. Construct a perpendicular from I to AB(segment), intersecting AB(segment) at a point R. 3. Using I as center and IR as radius, draw a circle.

Construction 10(Given a triangle, circumscribe a circle about the triangle.)

1. Construct the perpendicular bisectors of any two sides of triangle ABC. Label the point of inntersection O. 2. Using O as center and OA as radius, draw a circle.

Construction 9 (Given a point outside a circle, construct a tangent to the circle from the given point.)

1. Draw OP(line). 2. Find the midpoint, M, of OP by constructing the perependicular bisector of OP. 3. Using M as center and MP as radius, draw a circle thta intersects circle O in a point X. 4. Draw PX(ray).

Construction 8(Given a point on a circle, construct the tangent to the circle at the given point.)

1. Draw OP(ray). 2. Construct the line perpendicular to OP(ray)at P. Call it t.

Construction 14(Given two segments, construct their geometric mean.)

1. Draw a line and mark off RS=a and ST=b. 2. Locate the midpoint O of RT(segment) by constructing the perpendicular bisector of RT. 3. Using O as center draw a semicircle with a radius equal to OR. 4. At S, construct a perpendicular to RT. The perpendicular intersects the semicircle at a point Z. ZS is the geometric mean between a and b.

Construction 2 (Given an angle, construct an angle congruent to the given angle.)

1. Draw a ray. Label it RY. 2.Using B as center and any radius, darw an arc intersecting BA and BC(rays). Label the points of intersection D and E, respectively. 3. Using R as center and the same radius as in Steph 2, draw an arc intersecting RY(ray). Label the arc XS, with S the point wherer the arc intersects RY. 4. Using S as center and a radius equal to DE, draw an arc that intersects XS(arc) at a point Q. 5. Draw RQ(ray).

Construction 13(Given three segments, construct a fourth segment so that the four segments are in proportion.)

1. Draw an angle HIJ. 2. On IJ(ray), mark off IR=a and RS=b. 3. On IH(ray), mark off IT=c. 4. Draw RT(segment). 5. At S, construct a parallel to RT(segment), intersecting IH(ray) in a point U.

Comparing Area of Triangles(3 Parts)

1. If two triangles have equal heights, then the ratio of their areas equals the ratio of their basees. 2. If two triangles have equal bases, then the ratio of their areas eequals the ratio of their heights. 3. If two triangles are similar, theen the ratio of their areas equals the square of their scale factor.

Construction 7 (Given a point outside a line, construct the parallel to the given line through the given point.)

1. Let A and B be two points on line k. Draw PQ(line). 2. At P, construct angle 1 so that angle 1 and angle PQB are congruent corresponding angles. Let l be the line containing the ray you just constructed.

Two Principles of Geometric Probability

1. Suppose a point P of AB(segment) is picked at random. Then:probability that P is onn AC(segment)=length of AC/length of AB

Properties of a Regular Pyramids(4 of them)

1. The base is a regular polygon. 2. All lateral edges are conngruent. 3. All lateral faces are congruent isosceles triangles. The height of a lateral face is called the slant height of the pyramid. It is dennoted by l. 4. The altitude meets the base at its center, O.

Construction 1 (Given a segment, construct a segment congruent to the given segment.)

1. Use a straightedge to draw a linne. Call it l. 2. Choose any point on l and label it X. 3.Set your compass for radius AB. Using X as center, draw an arc intersecting lin l. Label the point of intersection Y.

Construction 3(Given an angle, construct the bisector of the angle.)

1. Using B as center and any radius, draw an arc that intersects BA(ray) at X and BC(ray) at Y. 2. Using X as center and a suitable radius, draw an arc. Using y as center and the same radius, draw an arc that intersects the arc with center X at a point Z. 3. Draw BZ(ray).

Construction 5 (Given a point on a line, construct the perpendicular to the line at the given point.)

1. Using C as center and any radius, draw arcs intersecting k at X and Y. 2. Using X as center and a radius greater than CX, draw an arc. Using y as center and the same radius, draw an arc intersecting the arc with center X at a point Z. 3. Draw CZ(line).

Construction 6(Given a point outside a line, construct the perpendicular to the line from the given point.)

1. Usinng R as center, draw two arcs of equal radii that intersect k at points X and Y. 2. Using X and Y as centers and a suitable radius, draw arcs that intersect at a point B. 3. Draw RB(line).

Image

The after in the before and after of the mapping.

Construction 4(Given a segment, construct the perpendicular bisector of the segment.)

1. Usinng any radius greater than AB/2, draw four arcs of equal radii, two with center A and two with center B. label the points of intersections of these arcs X and Y. 2. Draw XY(line).

Two Principles of Geometric Probability

2. Suppose a point P of region S is picked at ranndom. Then: probability that P is in region R=area of R/area of S.

Approximations for π

3.14,22/7,3.1416,3.14159

Measure of a Major Arc

360 minus the measure of its minor arc.

Formula for Measure of Ext. Angle

360/n

Diameter

A chord that containnes the center of a circle.

Composite

A combination of mappings is called a composite.

Corollary

A composite of reflections in perpendicular lines is a half turn about the point where the lines intersect.

Theorem 14-8

A composite of reflections in two intersecting lines is a rotation about the point of intersection of the two lines. The measure of the angle of rotation is twice the measure of the angle from the first line of reflection to the second.

Theorem 14-7

A composite of reflections in two parallel lines is a translation. The translation glides all points through twice the distance from the first line of reflection to the second.

Cone

A cone is like a pyramid except that its base is a circle instead of a polygon.

Counterclockwise and Clockwise Rotations

A counterclockwise rotation is considered positive, and a clockwise rotation is considered negative.

Cylinder

A cylinder is like a prism except that its bases are circles innstead of polygons.

Thm 9-5(Relationship between diameter that is perpendicular to chord)

A diameter that is perpendicular to a chord bisects the chord and its arc.

Corollary 3(How does the dilation chang ethe area of a polygon)

A dilation Do,k maps any polygon to a similar polygon whose area is k^2 times as large.

Corollary 2(How does a dilation change a parallel segment)

A dilation Do,k maps any segment to a parallel segment |k| times as long.

Dilation

A dilation is a transformation related to similarity rather than congruence.

Corollary 1(A dilation maps angle to a blank angle)

A dilation maps an angle to a congruent angle.

Theorem 14-5(What does a dilation do to a triangle)

A dilation maps any triangle to a similar triangle

Symmetry

A figure in the plane has symmetry if there is an isometry, other tha nthe identity, that mpas the figure onto itself.

Glide Reflection

A glide reflection is a transformation in which every point P is mapped to a point P" by these steps: 1. A glide maps P to P'. 2. A reflection in. a line parallel to the glide line maps P' to P".

Tangent

A line in the plane of a circle that intersects the circle in exactly one point, called the point of tangency.

Secant

A line that contains a chord.

Theorem 13-7 Slope-Intercept Form

A line with the equation y=mx+b has slope m and y-intercept b.

Linear Equation

A linear equation is an equation whose graph is a line.

Mapping

A mapping is a correspondence between sets of points.

Transformation

A one to one mapping from the whole plane to the whole planne is called a transformation.

One to one Mapping

A one to one mapping occurs if every member of B has exactly onne preimage in A.

Polygon inscribed inn a circle

A polygon is inscribed in a circle when each vertex of the polygon lies on nthe circle(polygon is inside of circle.)

Cube

A rectangular solid with square faces is a cube.

Theorem 14-2(What is a reflection)

A reflection in a line is an isometry.

Sector of a circle

A region bounded by two radii and an arc of the circle.

Rotation

A rotation about point O through x degrees is a transformation such that: 1. If a point P is different from O, then OP'=OP and measure of angle POP'=x. 2. If point P is the point O, then P'=P. Symbol for rotation is curvy R subcript O,degrees.

Half Turn

A rotation about point O throught 180 degrees is called a half turn about O and is usually denoted by H subscript O. Using coordinates, a half-turn Ho about the origin can be written H.:(x,y)-->(-x,-y)

Theorem 14-4(What is a rotation)

A rotation is an isometry

Chord

A segment whose endpoints lie on a circle.

Prism

A solid figure that has two congruent, parallel polygons as its bases. Its sides are parallelograms

Reflections

A transformation in a line acts like a mirror. A reflection in nline m maps every poinnt P to a point P' such that: 1. If P is not on the line m, then m is the perpendicular bisector of PP'(segment). 2. If P is onn line m(line of reflection/mirror line where the figure gets reflected), then P'=P.

Translation

A transformation that glides all points of the plane the same distance in the same direction is called a translation.

Theorem 14-3(What is a translation)

A translation is an isometry.

Area, A, of circle with radius r

A=πr²

Distance, Angle Measure, Orientation, and Area under a reflection

Distance, angle measure, and area are invariant under a reflection. However, the orientation of a figure is not invariant under a reflection because a reflection changes a clockwise orientation to a counterclockwise one.

Properties of Dilations

Do,k maps any poinnt P to a point P' determined as follows:1. If k>0, P' lies on OP(ray) and OP'=k*OP. 2. If k<0, P' lies on the ray opposite OP(ray) and OP'=|k|*OP. 3. The center O is its own image. If |k|>1, the dilation is called an expansion. If |k|<1, the dilation is called a contraction.

Thm 9-2(Converse of 9-1 but worded slightly differently)

If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.

Thm 9-1(Relationship between tangent and radius)

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.

9-7 Corollary 3(Opposite angles of a quadrilateral are ? if the quad. is inscribed in a circle)

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

Isometry

If a transformation maps every segment to a congruent segment, it is called an nisometry.

Thm 8-4(How to prove Acute Triangle)

If c^2<a^2+b^2, then measure of angle c<90, and triangle ABC is acute.

Thm 8-3( How to prove Right Triangle)

If c^2=a^2+b^2, then measure of angle c=90, and triangle ABC is right.

Thm 8-5(How to prove Obtuse Triangle)

If c^2>a^2+b^2, then measure of angle c>90, and triangle ABC is obtuse.

Right Prism

If the lateral faces of a prism are rectangles, the prism is a right prism.

Theorem 12-11(Three parts. Ratio of corresponding perimeters, base areas, lateral areas, and total areas, and volumes in similar solids)

If the scale factor of 2 similar solids is a:b, then (1)The ratio of the corresponding perimeters is a:b (2)The ratio of the base areas, of the lateral areas, and the total areas is a2: b2 (3)The ratio of the volumes is a3: b3

Theorem 11-7

If the scale factor of two similar figures is a:b, then(1)the ratio of the perimeters is a:b. (2)the ratio of the areas is a^2:b^2.

Thm 8-3(Converse of Pythagorean)

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Postulate 18 Area Congruence Postulate

If two figures are congruent, then they have the same area

9-7 Corollary 1 (Relationship between two inscribed angles that intercept the same arc)

If two inscribed angles intercept the same arc, then the angles are congruent

Thm 8-7(30-60-90 Theorem)

In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

Thm 8-6(45-45-90 Theorem)

In a 45-45-90 triangle, the hypotenuse is √2 times as long as a leg.

Altitude of a Right Cylinder

In a right cylinder, the segment joining the centers of the circular bases is an altitude.

Scalar Multiple of a Vector

In general, if the vector PQ=(a,b), then kPQ=(ka, kb); kPQ is called a scalar multiple of PQ. Multiplying a vector by a real number k multiplies the length of the vector by |k|. If k<0, the direction of the vector reverses as well.

Thm 9-4(Arcs have congruent blank, and other way around)

In the same circle or in congruent circles: (1)Congruent arcs have congruent chords. (2)congruent chords have congruent arcs.

Thm 9-6(Chords that are equally distant from center/s are blank, and other way around)

In the same circle or in congruent circles:(1)Chords equally distant from the center (or centers) are congruent. (2)Congrueent chords are equally distant from the center (or centers).

Thm 9-3 Arcs (congruent if)

In the same circles or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.

Altitude of a Prism

It is a segment joining the two base planes and perpendicular to both.

Length of Arc

Length of Arc=x/360 * 2πr

Circumference of a circle(words)

Like the perimeter

Identifying Slope and Direction

Lines with positive slope rise to the right. Lines with negative slop fall o the right. The greater the absolute value of a line's slope, the steeper the line. The slope of a horizontal line is zero. The slope of a vertical line is not defined.

Locus

Locus means a figure that is the set of all points, and only those points, that satisfy one or more conditions.

Right Cone

Note that "slant height" applies only to a regular pyramid and a right cone.

Oblique Prism

Otherwise, the prism is an oblique prism.

Orthocenter of a Triangle

Point where altitudes meet

Centroid of a Triangle

Point where medians meet

Sphere

Set of all points in space at a distance r from point O.

Similar Solids

Similar solids are solids that have the same shape but not necessarily the same size.(All spheres are similar)

3 Different Forms of Linear Equations

Standard form, slope-intercept form, and point-slope form.

Theorem 9-1 Corollary(Relationship between tangents from a point)

Tangents to a circle from a point are congruent.

Apothem of a regular polygon

The (perpendicular) distance from the center of the polygon to a side.