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Topic: Special right triangles

	Types Of Triangles
		A right triangle is a triangle with a right angle (i.e.
		The lengths of the sides of a right triangle are related by the Pythagorean Theorem.
		An acute triangle is a triangle whose angles are all acute (i.e.
	www.onlinemathlearning.com /types-of-triangles.html (725 words)

	Special Right Triangles
		A special right triangle is a right triangle whose sides are in a particular ratio.
		A 3-4-5 triangle is right triangle whose lengths are in the ratio of 3:4:5.
		A 45°- 45°- 90° triangle is a special right triangle whose angles are 45°, 45°and 90°.
	www.onlinemathlearning.com /special-right-triangles.html (732 words)

	GMAT Geometry Chapter 3: Triangles
		The height of the triangle is the perpendicular distance from the base to the opposite angle.
		There are also two special right triangles that we identify by the measurements of their angles.
		The height of an equilateral triangle always bisects the triangle, creating two equal right triangles that are both 30-60-90 right triangles.
	www.800score.com /content/guide7b2.html (1570 words)

	Relations and sizes - Right triangle facts - In Depth (Site not responding. Last check: 2007-10-12)
		This version of the right triangle is so popular that plastic models of them are manufactured and used by architects, engineers, carpenters, and graphic artists in their design and construction work.
		The ratio of this triangle's longest side to its shortest side is "two to one." That is, the longest side is twice as long as the shortest side.
		He proved that, for a right triangle, the sum of the squares of the two sides that join at a right angle equals the square of the third side.
	www.math.com /school/subject3/lessons/S3U3L4DP.html (505 words)

	Mere Math: "The Bare Facts" Triangle Trigonometry
		Because any triangle can be divided into several right triangles or a combination of right triangles, triangle trigonometry is the study of the properties of right triangles.
		Stated in words: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
		In the 30° - 60° - 90° right triangle, the smallest side is exactly ½ the length of the hypotenuse (or the hypotenuse is twice the length of the smallest side), and the length of the third side is √3 or ~ 1.7 times the length of the smallest side.
	polaris.umuc.edu /~mjohnso5/Trigonometry.html (701 words)

	Triangle - Wikipedia, the free encyclopedia
		A central theorem is the Pythagorean theorem stating that in any right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
		A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.
		The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base.
	en.wikipedia.org /wiki/Right_triangle (2243 words)

	Geometry: Special Triangles - Math for Morons Like Us
		It says - in a right triangle, the square of the measure of the hypotenuse equals the sum of the squares of the measures of the two legs.
		One of the special right triangles which we deal with in geometry is an isosceles right triangle.
		In a 45-45-90 triangle, the measure of the hypotenuse is equal to the measure of a leg multiplied by SQRT(2).
	library.thinkquest.org /20991/geo/stri.html (715 words)

	edHelper.com - High School Geometry - Right Triangles
		Triangle ABC is a 45°-45°-90° triangle with C at the right angle.
		In right triangle ABC, the length of leg BC is 4
		In triangle ABC, the ratio of the measure of angle A to angle B is 1:3.
	www.edhelper.com /math/geometry_right_triangles3.htm (174 words)

	SparkNotes: Special Triangles: Right Triangles
		The total angle sum of a triangle is 180 degrees, and the right angle is 90 degrees, so the other two must sum to 90 degrees.
		One is the right triangle formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles.
		One last characteristic to note is that the legs of a right triangle are also altitudes of the triangle.
	www.sparknotes.com /math/geometry2/specialtriangles/section4.rhtml (340 words)

	KLRN TeacherLine - Lesson Plans
		The length of the hypotenuse of a 30-60-90 triangle is 20m.
		The measure of the angles of a triangle are 3n, 3n, and 6n.
		In a 30-60-90 triangle, the length opposite the 60° is 9.
	www.klrn.org /Teacherline/lesson_plans/special.html (954 words)

	Discover the Wisdom of Mankind on right triangles (Site not responding. Last check: 2007-10-12)
		This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the thirdandmdash;something unique to right triangles.
		The area S of a triangle is Sandnbsp;=andnbsp;andfrac12;bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base.
		In three dimensions the area of a general triangle {Aandnbsp;=andnbsp;(x1,andnbsp;y1,andnbsp;z1), Bandnbsp;=andnbsp;(x2,andnbsp;y2,andnbsp;z2) and Candnbsp;=andnbsp;(x3,andnbsp;y3,andnbsp;z3)} is the 'pytagorean' sum of the area's of the respective projections on the three principal planes (i.e.
	www.blinkbits.com /blinks/right_triangles (2774 words)

	Topics in trigonometry: The 30°-60°-90° triangle
		Since this is a right triangle and angle A is 60°, then the remaining angle B is its complement, 30°.
		In the right triangle DEF, angle D is 30°, and side DF is 3 inches.
		Hence AO, OB are radii, and triangle AOB is isosceles.
	www.themathpage.com /aTrig/30-60-90-triangle.htm (993 words)

	Acute Angles in ``Special'' Right Triangles
		triangle, the length of the hypotenuse is twice that of the side opposite the 30 degree angle, and the side adjacent to the 30 degree angle is
		These relationships, when applied to the right triangle definitions of the trig functions, allow us to easily find and express the exact value of any trig function of such an angle.
		A consequence of these special angle relationships is, given one trigonometric functional value of such an angle, the other five are easily determined from the right triangle definitions and the Pythagorean Theorem.
	aah.ryan-usa.com /node73.html (168 words)

	Special Right Triangles (Site not responding. Last check: 2007-10-12)
		This isosceles right triangle is important because it is half of a square.
		The other important right triangle to memorize is the 30-60-90 right triangle.
		This triangle is important because it is half of an equilateral triangle, and allows us to find the height of an equilateral triangle.
	www.dean.tec.ma.us /MCAS/mcas-srt.htm (327 words)

	GeoWeb - Special Right Triangles
		Students will be able to identify the special right triangles 45-45-90 and 30-60-90.
		i) In a 30-60-90 right triangle, the measure of the hypotenuse is two times that of the leg opposite the 30-degree angle.
		Suggestions/Comments: A worksheet comprised of a lot of 45-45-90 and 30-60-90 triangles in solving for angles and side measurements would be a good way for students to grow accustom to using the new properties.
	teacherlink.org /content/math/interactive/geoweb/lessons/sprttri/home.html (400 words)

	Geometry: Congruent Right Triangles - Math for Morons Like Us
		Right triangles are special triangles that contain one right angle.
		Since there is one leg and one acute angle in each triangle that is congruent to another leg and another acute angle in the other triangle, both of which are right, triangle JKL is congruent to triangle MNO by the Leg-Acute Angle Theorem.
		It states if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
	library.thinkquest.org /20991/geo/crtri.html (675 words)

	Special Triangles
		In the sense that trigonometry is the computational side of geometry, the theorem about the sum of the angles of a triangle and the pythagorean theorem are the beginnings of trigonometry.
		Geometry goes one step further into becoming trigonometry by looking at certain special right triangles where by learning the ratios of the sides we can compute the lengths of all of the sides by knowing just one side.
		If, like before, we make the hypotenuse (also the side of the equilateral triangle) be 1, then the short side will be 1/2, because it is half the side length of the equilateral triangle.
	www.cgtcollege.org /mat104/special_triangles.htm (328 words)

	Math Forum - Problems Library - Geometry - Special Right Triangles
		A square with side length s is inscribed in an equilateral triangle of side length t.
		A chord of a circle is the hypotenuse of an isosceles right triangle whose legs are radii of the circle.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/problems/sets/geo_special_rttri.html (541 words)

	Special Right Triangles
		An Isosceles right triangle is a special triangle with several special properties.
		The right triangle, which is half of an equilateral (equiangular) triangle, has special properties also.
		triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is
	jwilson.coe.uga.edu /EMT668/EMAT6680.Folders/Brooks/6690stuff/Righttriangle/SpecialRight.html (219 words)

	Day1
		We have been using the Pythagorean Theorem to find missing values of the sides of right triangles.
		There are six problems so as long as they attempt as of the problems and get 4 out of the 6 right they will receive a passing homework grade.
		We will review the Pythagorean theorem, ("What is Formula for the Pythagorean Theorem?") and both types of Special Right Triangles ("What is the side length ratio of a 45° - 45° - 90° triangle?" "What is the side length ratio for a 30° - 60° - 90° triangle?").
	home.olemiss.edu /~ksbost/day1.htm (331 words)

	Right Triangles (Site not responding. Last check: 2007-10-12)
		Note: as usual, in all exercises on right triangles c stands for the hypotenuse, a and b for the perpendicular sides, and A and B for the angles opposite to a and b respectively.
		In each of the following right triangles of which two sides are given, compute the sin, cos, and tan of the angles A and B.
		One side of a right triangle is r, the radius of the earth, and the hypotenuse is r + h where h is the height of the lighthouse.
	aleph0.clarku.edu /~djoyce/java/trig/right.html (2174 words)

	Glencoe Mathematics - Online Study Tools
		The measures of both legs of a right triangle are 4.
		Which of the following special right triangles can have a Pythagorean triple as the lengths of its sides?
		neither a 45°-45°-90° triangle nor a 30°-60°-90° triangle
	www.glencoe.com /sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-02-825275-6&chapter=8&lesson=2 (65 words)

	Fractal Triangles: Complements to Basic and Special Right Triangles by Charles William Johnson (E-book) in (Site not responding. Last check: 2007-10-12)
		Fractal Triangles presents different series of triangles that are complementary to the basic and special triangles in forming a circle.
		This complementarity of right triangles reveals the significance of the fractal right triangles.
		Fractal Triangles discusses the possible relationship of the right triangle to the physical and chemical constants, as well as other relationships of spacetime/motion (matter-energy), and touches on the union of geometry with chemistry and physics.
	www.lulu.com /content/385836 (191 words)

	Basic Triangle Values
		These are the triangles that will typically be used in math problems, because you can get an exact answer (using square roots), instead of being stuck with a decimal approximation.
		-degree triangle at the right, using the given values for the lengths of the sides:
		(Note that it is irrelevant what are the lengths of the actual triangle you are dealing with; this reference triangle gives you the ratios, or the trigonometric values.) The "theta" (the circle with the line through it) in the lower left corner is the angle we'll use.
	www.purplemath.com /modules/trig.htm (583 words)

	Special Right Triangles
		Each of the two triangles that make up the square have equal lengths of the legs (because all sides of a square are equal).
		They also each have a 90 degree angle and two 45 degree angles (which goes along with the fact that the legs are equal).
		In an isosceles right triangle, if the legs have length a, then the hypotenuse will have a length of a√2.
	rachel5nj.tripod.com /NOTC/specialrighttriangles.html (137 words)

	Geometric Means of Right Triangles and their altitudes
		For a right triangle and an altitude drawn from the vertex of the right angle to the hypotenuse (as shown in the picture in the top of the page)
		The altitude is the the geometric mean between the measures of the two segments of the hypotenuse
		The measure of the leg of the triangle is the geometric mean between the measure of the hypotenuse and the measure of the segment of the hypotenuse adjacent to that leg.
	www.wccusd.k12.ca.us /hercmh/MWT/RightTriangleSpecialProperties1.htm (260 words)

	Day1
		The student will be able to explain examples of real-life instances that have properties of Special Right Triangles.
		Discuss the two real-life instances in the textbook where properties of Special Right Triangles can be used.
		Discuss with the students the objects in the classroom that have properties of Special Right Triangles.
	home.olemiss.edu /~ksbost/day2.htm (478 words)

	Cool math .com - The Geometry of Triangles
		Definitions and formulas for the area of a triangle, the sum of the angles of a triangle, the Pythagorean theorem, Pythagorean triples and special triangles (the 30-60-90 triangle and the 45-45-90 triangle)
		The sides, a and b, of a right triangle are called the legs, and the side that is opposite to the right (90 degree) angle, c, is called the hypotenuse.
		This formula will help you find the length of either a, b or c, if you are given the lengths of the other two.
	www.coolmath.com /reference/triangles-geometry.html (187 words)

	Special Right Triangles
		There are two "special" formulas that apply ONLY to the 45º-45º-90º triangle.
		A nice feature of these special formulas is that the answer is already in reduced form
		Since a 45º-45º-90º, also called an isosceles right triangle, has two legs equal, we know that the other leg also has a length of 7 units.
	regentsprep.org /Regents/math/rtritrig/Ltri45.htm (178 words)

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