Obj. 24 Special Right Triangles
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Identify when a triangle is a 45-45-90 or 30-60-90 triangle Use special right triangle relationships to solve problems
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4.11.1 Pythagorean Theorem
The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to find missing sides of right triangles and to classify triangles as right, obtuse, or acute based on side lengths. It also introduces Pythagorean triples, which are sets of integers that satisfy the Pythagorean theorem.
4.11.2 Special Right Triangles
1) Special right triangles have properties that allow for determining side lengths without using the Pythagorean theorem. The 45-45-90 triangle has legs that are equal in length and a hypotenuse that is √2 times the leg length. The 30-60-90 triangle has a shorter leg that is half the hypotenuse, a longer leg that is √3 times the shorter leg, and a hypotenuse that is 2 times the shorter leg.
2) Examples demonstrate using the properties of special right triangles to find missing side lengths by setting up and solving equations based on the corresponding theorems. Rationalizing denominators may be required when finding a leg from a known hypotenuse.
3)
Special right triangles
This document discusses using shortcuts and properties of right triangles to solve for unknown side lengths. It explains that right isosceles triangles have two congruent legs and a 45-45-90 angle relationship. The shortcut for these triangles is that the hypotenuse is equal to one leg times the square root of 2. It also discusses 30-60-90 triangles having a short leg opposite the 30 degree angle, a hypotenuse twice as long as the short leg, and a long leg equal to the short leg times the square root of 3. Several examples are provided to demonstrate using these properties and shortcuts to find missing side lengths of right triangles.
Special Right Triangles
The document discusses two types of special right triangles - 45-45-90 triangles and 30-60-90 triangles. For 45-45-90 triangles, the ratio of sides is 1-1-√2 and the diagonal of a square forms two 45-45-90 triangles. For 30-60-90 triangles, the ratio of sides is 1-√3-2 and the altitude of an equilateral triangle forms two 30-60-90 triangles. Examples are given of calculating missing side lengths using properties of these special right triangles.
Special Right Triangles
1) Special right triangles have specific angle measurements (30-60-90 or 45-45-90) that result in consistent side length ratios.
2) The Pythagorean theorem, a2 + b2 = c2, always applies to right triangles and relates the sides.
3) Key properties of 30-60-90 and 45-45-90 triangles include specific ratios between short, medium, and long sides that remain consistent regardless of the triangle's size.
Obj. 24 Special Right Triangles
Identify when a triangle is a 45-45-90 or 30-60-90 triangle
Use special right triangle relationships to solve problems
4.11.2 Special Right Triangles
This document discusses special right triangles and their properties. It defines 45-45-90 and 30-60-90 triangles, and provides the key relationships between their sides: for 45-45-90 triangles, the hypotenuse is √2 times the leg; for 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg. It provides examples of using these relationships to solve for missing side lengths.
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The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to find missing sides of right triangles and to classify triangles as right, obtuse, or acute based on side lengths. It also introduces Pythagorean triples, which are sets of integers that satisfy the Pythagorean theorem.
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The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to find missing sides of right triangles and to classify triangles as right, obtuse, or acute based on side lengths. It also introduces Pythagorean triples, which are sets of integers that satisfy the Pythagorean theorem.
4.11.2 Special Right Triangles
1) Special right triangles have properties that allow for determining side lengths without using the Pythagorean theorem. The 45-45-90 triangle has legs that are equal in length and a hypotenuse that is √2 times the leg length. The 30-60-90 triangle has a shorter leg that is half the hypotenuse, a longer leg that is √3 times the shorter leg, and a hypotenuse that is 2 times the shorter leg.
2) Examples demonstrate using the properties of special right triangles to find missing side lengths by setting up and solving equations based on the corresponding theorems. Rationalizing denominators may be required when finding a leg from a known hypotenuse.
3)
Special right triangles
This document discusses using shortcuts and properties of right triangles to solve for unknown side lengths. It explains that right isosceles triangles have two congruent legs and a 45-45-90 angle relationship. The shortcut for these triangles is that the hypotenuse is equal to one leg times the square root of 2. It also discusses 30-60-90 triangles having a short leg opposite the 30 degree angle, a hypotenuse twice as long as the short leg, and a long leg equal to the short leg times the square root of 3. Several examples are provided to demonstrate using these properties and shortcuts to find missing side lengths of right triangles.
Special Right Triangles
The document discusses two types of special right triangles - 45-45-90 triangles and 30-60-90 triangles. For 45-45-90 triangles, the ratio of sides is 1-1-√2 and the diagonal of a square forms two 45-45-90 triangles. For 30-60-90 triangles, the ratio of sides is 1-√3-2 and the altitude of an equilateral triangle forms two 30-60-90 triangles. Examples are given of calculating missing side lengths using properties of these special right triangles.
Special Right Triangles
1) Special right triangles have specific angle measurements (30-60-90 or 45-45-90) that result in consistent side length ratios.
2) The Pythagorean theorem, a2 + b2 = c2, always applies to right triangles and relates the sides.
3) Key properties of 30-60-90 and 45-45-90 triangles include specific ratios between short, medium, and long sides that remain consistent regardless of the triangle's size.
Obj. 24 Special Right Triangles
Identify when a triangle is a 45-45-90 or 30-60-90 triangle
Use special right triangle relationships to solve problems
4.11.2 Special Right Triangles
This document discusses special right triangles and their properties. It defines 45-45-90 and 30-60-90 triangles, and provides the key relationships between their sides: for 45-45-90 triangles, the hypotenuse is √2 times the leg; for 30-60-90 triangles, the hypotenuse is 2 times the shorter leg and √3 times the longer leg. It provides examples of using these relationships to solve for missing side lengths.
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The document discusses the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It provides examples of using the theorem to find missing sides of right triangles and to classify triangles as right, obtuse, or acute based on side lengths. It also introduces Pythagorean triples, which are sets of integers that satisfy the Pythagorean theorem.
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This document discusses 30-60-90 triangles and provides shortcuts for determining side lengths. It states that in a 30-60-90 triangle: (1) the side opposite the 30 degree angle is the short side s, the side opposite the 60 degree angle is the long side l, and the hypotenuse is h; (2) the relationship between s and h is h = 2s; and (3) the relationship between s and l is l = s√3. It then demonstrates using these shortcuts to find side lengths when given s, l, or h through worked examples.
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This document provides a study package on circles for a mathematics class. It begins with an index listing the topics covered, which include theory, revision, exercises, assertion and reason questions, and past examination questions. It then covers circle theory, equations of circles in various forms including parametric and Cartesian, intercepts made by circles on axes, the position of points with respect to circles, lines and circles, and tangents to circles. Examples are provided to illustrate each concept. The document is intended to be a comprehensive resource for students to learn about circles.
Chapter 5 unit 9
The Pythagorean theorem is used to calculate the length of the hypotenuse of a right triangle using the lengths of the other two sides. It states that the sum of the squares of the two legs equals the square of the hypotenuse. The distance formula calculates the distance between two points and involves taking the difference of the x-coordinates squared and the difference of the y-coordinates squared and adding those values together. It is used to find distances in applications like navigation and physics.
Application of Integration
The document discusses applications of integration, including calculating the length of a curve and surface area of solids obtained by rotating curves. It provides formulas for finding the arc length of a curve given by y=f(x), and surface area of solids obtained by rotating curves about the x- or y-axes. Examples are worked out applying these formulas to find the arc length of curves and surface area of rotated regions. The document also discusses evaluating triple integrals to find the volume of a three-dimensional region and using triple integrals to find the centroid of a volume.
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6.14.4 Angle Relationships
1) When a tangent and secant or chord intersect at the point of tangency, the measure of the angle formed is half the measure of the intercepted arc.
2) When two secants or chords intersect in the interior of a circle, the measure of each angle formed is half the sum of the intercepted arcs.
3) When secants or tangents intersect outside a circle, the measure of the angle formed is half the difference between the intercepted arcs. Algebraic formulas are provided to calculate angle measures using the intercepted arcs.
8 4 Special Rt Triangles Noted
This document discusses two types of special right triangles: 45-45-90 triangles and 30-60-90 triangles. For 45-45-90 triangles, the hypotenuse is √2 times as long as each leg. For 30-60-90 triangles, the hypotenuse is twice as long as the shorter leg and the longer leg is √3 times as long as the shorter leg. It provides examples of using these properties to find missing side lengths.
Applications of Differential Calculus in real life
Applications of Differentiation in economics, optimization, kinematics, tangent and normal lines, related rates, linear approximation or small changes
circles_ppt angle and their relationship.ppt
The document provides information about properties of circles, including theorems about angles formed by chords, secants, and tangents intersecting inside and outside circles, as well as theorems about relationships between lengths of segments of chords and secants. It also discusses writing equations of circles in standard form given the center and radius, finding the center and radius from a standard equation, and graphing circles from standard equations. Examples are provided to demonstrate applying the theorems and writing/graphing circle equations.
Pythagorean theorem
The document discusses the Pythagorean theorem and provides examples of using it to solve for unknown sides of right triangles. It defines the theorem as: for a right triangle with sides a, b, and hypotenuse c, c^2 = a^2 + b^2. Several practice problems are shown applying the theorem to find missing lengths. Solutions to challenging problems involving using the theorem to find perimeters and areas are also provided.
Section 10.2
The document discusses calculus concepts related to parametric curves, including:
1) Finding the tangent line to a parametric curve using derivatives.
2) Computing the area under a parametric curve using integrals.
3) Calculating the arc length of a parametric curve using integrals.
4) Determining the surface area of a surface of revolution generated by rotating a parametric curve about an axis.
Chapter 5 Using Intercepts
This document provides examples and explanations for finding the x-intercept and y-intercept of linear equations and using those intercepts to graph the line. It discusses interpreting the intercepts in real world contexts like time to complete a race or amount of items that can be purchased. Students are given examples of finding intercepts, graphing lines by plotting the intercept points, and word problems involving linear equations.
Chapter 5 Using Intercepts
This document provides examples and explanations for finding the x-intercept and y-intercept of linear equations and using those intercepts to graph the line. It discusses interpreting the intercepts in real world contexts like time to complete a race or amount of items that can be purchased. Students are given examples of finding intercepts, graphing lines by plotting the intercept points, and word problems involving linear equations.
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* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
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* Use the vertical line test to determine a function
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This document discusses functions and their graphs. It defines increasing, decreasing and constant functions based on how the function values change as the input increases. Relative maxima and minima are points where a function changes from increasing to decreasing. Symmetry of functions is classified by the y-axis, x-axis and origin. Even functions are symmetric about the y-axis, odd functions are symmetric about the origin. Piecewise functions have different definitions over different intervals.
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This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
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Obj. 24 Special Right Triangles
- 1. Obj. 24 Special Right Triangles The student is able to (I can): • Identify when a triangle is a 45-45-90 or 30-60-90 triangle • Use special right triangle relationships to solve problems
- 2. Consider the following triangle: To find x, we would use a2 + b2 = c2, which gives us: What would x be if each leg was 2? 1 1 x 2 2 2 2 1 1 x x 1 1 2 x 2 + = = + = =
- 3. Again, we will use the Pythagorean Theorem Simplifying the radical, we can factor to give us Do you notice a pattern? 2 2 x 2 2 2 2 2 2 x x 4 4 8 x 8 + = = + = = 8 2 2.
- 4. Thm 5-8-1 45º-45º-90º Triangle Theorem In a 45º-45º-90º triangle, both legs are congruent, and the length of the hypotenuse is times the length of the leg. 2 x x x 245º 45º
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- 6. If we know the hypotenuse and need to find the leg of a 45-45-90 triangle, we have to divide by . This means we will have to rationalize the denominator, which means to multiply the top and bottom by the radical. The shortcut for this is to divide the hypotenuse by 2 and then multiply by 2 16 x 16 16 2 x 2 2 2 = = 16 2 8 2 2 = = 2. 16 x 2 8 2 2 = =
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- 8. Thm 5-8-2 30º-60º-90º Triangle Theorem In a 30º-60º-90º triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is times the length of the shorter leg. Note: the shorter leg is always opposite the 30º angle; the longer leg is always opposite the 60º angle. 3 x 2xx 3 60º 30º
- 9. Examples Find the value of x. Simplify radicals. 1. 2. 3. 4. 7 x 60º 30º 11x 9 x 60º 1616 60º x 9 3 16 3 8 3 2 = 14 11 5.5 2 =
- 10. Examples To find the shorter leg from the longer leg: Find the value of x 1. 2. 9 x 60º 10 x 30º longer leg 3 longer leg 3 3 3 3 = 9 x 3 3 3 3 = = 10 x 3 3 =