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Geometry

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		Triangle types ^Votes:0 Types of Triangles Right triangle: Has one 90 degree angle Equilateral triangle: All angles are the same (60 degrees) Isosceles triangle: Has two angles the same and two sides the same Scalene triangle: Has all three angles and all three sides different Obtuse triangle: Has one obtuse angle, greater than 90 degrees The sum of all angles in a triangle is ___________ degrees. Read More Go to Site


		Angles and Triangles in a Circle ^Votes:0 www.picciotto.org/math-ed/angles Adapted from Geometry Labs , by Henri Picciotto (Key Curriculum Press) Angles and Triangles in a Circle Equipment: Circle Geoboard, Circle Geoboard Paper Types of triangles: Equilateral (EQ) Acute Isosceles (AI) Right Isosceles (RI) Obtuse Isosceles (OI) Acute Scalene (AS) Right Scalene (RS) Half-Equilateral (HE) Obtuse Scalene (OS) Make triangles on the circle geoboard, with one vertex at the center and the other two on the circle. (See circle 1 above.) Make one of each of the eight types of triangles listed above, if possible. You do not have to do them in order! Sketch one of each of the types of triangles it was possible to make on circle geoboard paper. Identify which type of triangle it is, and label all three of its angles with their measures in degr Read More Go to Site

		Area and perimeter ^Votes:0 Area and perimeter Area Area of a square Area of a rectangle Area of a parallelogram Area of a trapezoid Area of a triangle Area of a circle Perimeter Circumference of a circle Math Contests School League Competitions Contest Problem Books Challenging, fun math practice Educational Software Comprehensive Learning Tools Visit the Math League Area The area of a figure measures the size of the region enclosed by the figure. This is usually expressed in terms of some square unit. A few examples of the units used are square meters, square centimeters, square inches, or square kilometers. Area of a Square If l is the side-length of a square, the area of the square is l 2 or l × l . Example: What is the area of a square having side-length 3.4? The area is the square of the side-length, which is 3 Read More Go to Site

		Area Formula Lab ^Votes:0 Printout Instructions for the Area Formula Lab: Go to each page below to print the activity OR use the "print all linked documents" if your browser has that capability. Cut out all "cut out pages" before you start the activity. Follow the activities in order 1 through 6. Activities 1. Squares and Rectangles 2. Parallelograms 3. Triangles 4. Trapezoid 5. Extra Figures 6. Area of a Circle (Cut Out Page) 7. a. Triangles and Parallelograms (Cut Out Page) 7. b. Trapezoids (Cut Out Page) 8. Extra Grid Paper (optional) Top Copyright 1998 by Margo Lynn Mankus Read More Go to Site

		Areas, Volumes, Surface Areas ^Votes:0 [text:Areas, Volumes, Surface Areas ] ( pi = [pi] = 3.141592...) [text:Areas] [text:square] = a^{2} [text:rectangle] = ab [text:parallelogram] = bh [text:trapezoid] = h/2 (b 1 + b 2 ) [text:circle] = pi r 2 [text:ellipse] = pi r 1 r 2 [text:triangle] = (1/2) b h [text:equilateral triangle] = (1/4) (3) a^{2} [text:triangle given SAS] = (1/2) a b sin C [text:triangle given a,b,c] = [sqrt][s(s-a)(s-b)(s-c)] [text:when] s = (a+b+c)/2 ([text:Heron's formula]) [text:regular polygon] = (1/2) n sin(360°/n) S^{2} [text:when n = # of sides and S = length from center to a corner] [text:Volumes] [text:cube] = a^{3} [text:rectangular prism] = a b c [text:irregular prism] = b h [text:cylinder] = b h = [pi] r^{2} h [text:pyramid] = (1/3) b h [text:cone] = (1/3) b h = 1/3 [pi] r^{2} h [text:sphere] = (4/3 Read More Go to Site

		Circles ^Votes:0 Circles a circle Definition: A circle is the locus of all points equidistant from a central point. Definitions Related to Circles arc: a curved line that is part of the circumference of a circle chord: a line segment within a circle that touches 2 points on the circle. circumference: the distance around the circle. diameter: the longest distance from one end of a circle to the other. origin: the center of the circle pi ( ): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle. radius: distance from center of circle to any point on it. sector: is like a slice of pie (a circle wedge). tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle. diameter = 2 x radius of circle Circumference of Circle = PI x diameter = 2 PI x ra Read More Go to Site

		Geometry ^Votes:0 Sorted by Grade Level Study Basic Math Skills Unlimited Interactive Practice Explanations and Examples Challenge Games Hundreds of Pages Kindergarten \| First \| Second \| Third \| Fourth \| Fifth \| Sixth \| Seventh \| Eighth Contact AAAMath Buy the AAAMath CD Sorted by Subject Addition Algebra Comparing Counting Decimals Division Equations Estimation Exponents Fractions Geometry Graphs Measurement Mental Math Money Multiplication Names of Numbers Patterns Percent Place Value Practical Math Properties Proportions Statistics Subtraction Please tell others about this site. Geometry - Table of Contents Geometry - Topics Geometry Facts and Calculations Area Perimeter and Circumference Surface Area Volume Return to Top Geometry - Lessons Geometry Facts and Calculations Polygons Polygons II Classifying Read More Go to Site

		Geometry2.html ^Votes:0 ©GLG 1998 HOME Read More Go to Site

		Geometry3.html ^Votes:0 ©GLG 1998 HOME Read More Go to Site

		Math Forum - Ask Dr. Math ^Votes:0 Associated Topics \|\| Dr. Math Home \|\| Search Dr. Math What is Factoring? Date: 01/17/98 at 14:01:47 From: Melissa Subject: Factoring I do not get how to do this algebra. One of the sample questions is: x squared + 5x + 4 I don't understand how to start out factoring. Date: 01/20/98 at 21:15:47 From: Doctor Loni Subject: Re: Factoring Factoring can be a very intimidating thing, but once you get the hang of it you will be a pro. First, let me make sure you understand what factoring is. To factor anything, you get it into its smaller component parts, finding out what smaller parts make up the bigger part. For instance, when you factor a number, say 12, you break it into its smallest parts. You could start with 6 x 2. Then 6 can be factored into 2 x 3, so your factors of 12 could be 2 x 2 x 3. Read More Go to Site

		Math Forum: Ask Dr. Math FAQ: Triangle Formulas ^Votes:0 --> Ask Dr. Math: FAQ T riangle F ormulas Geometric Formulas: Contents \|\| Ask Dr. Math \|\| Dr. Math FAQ \|\| Search Dr. Math also see Defining Geometric Figures Triangle A polygon (plane figure) with 3 angles and 3 sides. Sides: a, b, c Opposite angles: A, B, C Altitudes: h a , h b , h c Medians: m a , m b , m c Angle bisectors: t a , t b , t c Perimeter: P Semiperimeter: s Area: K Radius of circumscribed circle: R Radius of inscribed circle: r To read about triangles, visit The Geometry Center. Equilateral Triangle A triangle with all three sides of equal length. a = b = c. A = B = C = Pi/3 radians = 60 o P = 3a s = 3a/2 K = a 2 sqrt(3)/4 h a = m a = t a = a sqrt(3)/2 R = a sqrt(3)/3 r = a sqrt(3)/6 JavaSketchpad exploration: Equilateral triangle Isosceles Triangle A triangle with two sides Read More Go to Site

		section10 ^Votes:0 QUADRILATERALS: DEFINITION OF A QUADRILATERAL, PARALLELOGRAMS. The next types of geometric objects that we are going to consider are called quadrilaterals . A quadrilateral is a figure in a plane formed by connecting four segments endpoint to endpoint with each segment intersecting exactly two others. Each segment is called a side of the quadrilateral . Each endpoint where the sides meet is called a vertex of the quadrilateral . A quadrilateral is usually named after its vertices e.g. a quadrilateral ABCD . If two vertices of a quadrilateral are connected by a side, then they are called consecutive vertices . Otherwise they are called nonconsecutive vertices . If two sides of a quadrilateral share a common vertex, then they are called consecutive sides . Otherwise they are called nonconsec Read More Go to Site

		section11 ^Votes:0 PROPERTIES OF PARALLELOGRAMS, A RECTANGLE, A SQUARE AND A RHOMBUS. The following statement gives the main property of parallelograms. Theorem 11.1 If ABCD is a parallelogram then its nonconsecutive sides and its nonconsecutive angles are equal. Proof We need to prove that AB = CD, BC = AD , Let O be the point of intersection of diagonals AC and BD . Triangles BOC and DOA are equal by SAS : angles BOC and DOA are equal as vertical angles, BO = OD and AO = OC by theorem 10.1*. Therefore AD = BC . The equality of AB and CD is proved in the same way from the equality of triangles DOC and AOB . Triangles BAD and BCD are equal by SSS : they share a common side BD , AB = CD and AD = BC from the reasoning above. Therefore angles A and C are equal. The equality of angles B and D follows from the eq Read More Go to Site

		section12 ^Votes:0 ANGLES AND PARALLEL LINES, TRIANGLE'S MIDSEGMENTS. In order to investigate the further properties of quadrilaterals first we have to discuss some results involving angles and parallel lines. Theorem 12.1 Consider an angle and parallel lines that intersect with the angle. If the parallel lines divide one side of the angle into equal parts then they divide the other side of the angle into equal parts as well. Proof Let's consider an angle P1BQ1 and three parallel lines m1 , m2 , and m3 that intersect with P1BQ1 . Suppose that P1P2 = P2P3 then we need to show that Q1Q2 = Q2Q3 . Let EF be a line that passes through Q2 and is parallel to P1B . Then we have two parallelograms: P3FQ2P2 and P2Q2EP1. From the theorem 11.1 it follows that P2P3 = FQ2 and P1P2 = Q2E . But P1P2 = P2P3 by the assumption Read More Go to Site

		section5 ^Votes:0 VARIOUS TYPES OF TRIANGLES: ISOSCELES AND EQUILATERAL TRIANGLES. In this section we are going to continue our discussion of triangles. We start with the considering of various types of triangles. A triangle is called an isosceles triangle if it has two sides with equal lengths. Consider an isosceles triangle ABC with AC=CB . Then the side AB (i.e. the side that is not equal to the other sides) is called the base side . The angle opposite the base side is called the vertex angle and the rest two angles of an isosceles triangle are called base angles . Theorem 5.1 Suppose that ABC is an isosceles triangle then its base angles are equal. Proof We are given the fact that ABC is an isosceles triangle e.g. we have that AC=CB . Then we need to prove that But the triangle CAB is equal to the trian Read More Go to Site

		section6 ^Votes:0 ANGLES IN A TRIANGLE, RIGHT TRIANGLES AND EQUALITY OF RIGHT TRIANGLES BY HYPOTENUSE-LEG, EQUALITY OF TWO TRIANGLES BY SIDE-SIDE-SIDE. In this section we are going to finish our discussion of triangles and their properties. One of the remarkable results regarding angles in any triangle is formulated in the following theorem. Theorem 6.1 Sum of degree measures of all angles in a triangle is equal to 180 degrees. Proof Let ABC be an arbitrary triangle . We need to prove that Let's construct a line p such that it will pass through the point C and will be parallel to AB . Let D be a point on p located right to the point C . By construction angles DCB and CBA are equal as alternate interior angles for parallel lines p and AB and their transversal CB . At the same time angles DCA and CAB are one- Read More Go to Site

		section7 ^Votes:0 CIRCLES: DEFINITION OF A CIRCLE, CHORDS, TANGENT AND SECANT LINES. This section introduces the properties of the circles . We start with the definition of this geometric object. A circle is the set of all points in a plane at a fixed distance from a fixed point in the plane. The fixed point is called the center of the circle . The fixed distance is called the radius of the circle . A segment from a point of the circle to the center is also called a radius . Usually a circle is named by its center e.g. circle O . Sometimes we are also going to specify the radius r of the circle and name the circle as (O, r). We say that two circles are congruent if they have the same radius. If two or more circles share the same center then they are called concentric circles . A segment that connects two di Read More Go to Site

		section8 ^Votes:0 CIRCLES AND ANGLES. Now we are going to discuss the possible relationships between circles and angles. We start with the following definition. An angle ABC is called an angle inscribed in a circle O if the vertex B is a point on the circle O and segments AB and AC are chords of O . An angle OAC is called a central angle of a circle O if A and C are both points on the circle O . Theorem 8.1 Suppose that O is a circle and an angle ABC is an angle inscribed in O . Then Proof We need to show that an inscribed angle is equal to the half of the corresponding central angle. In order to do this we have to consider all possible cases that may happen for an inscribed angle. Case 1: One of the angle sides passes through O . Let's denote the degree measure of ABC as x . Then we need to show that the d Read More Go to Site

		section9 ^Votes:0 CIRCLES AND TRIANGLES. Another big topic for our discussion is given to us by possible relationships between circles and triangles. But first we have to introduce a few auxiliary definitions. We are going to say that lines (segments or rays) are called concurrent if they lie in the same plane and all intersect in the same point. The point of intersection is called the point of concurrency. Consider an arbitrary triangle ABC . The point of concurrency of the three angle bisectors in the triangle ABC is called the incenter . . The point of concurrency of the three lines, perpendicular to each side of the triangle ABC and passing through the corresponding middle point is called the circumcenter . Finally, the point of concurrency of the three altitudes or three lines through the altitudes of Read More Go to Site

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