Curriculum Geometry B/ Algebra II
UNIT 5: GEOMETRIC AND ALGEBRAIC CONNECTIONS
APPLY GEOMETRIC CONCEPTS IN MODELING SITUATIONS |
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MGSE9-12.G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
MGSE9-12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
MGSE9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
MGSE9-12.G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
MGSE9-12.G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
KEY IDEAS
- Modeling can be applied to describe real-life objects with geometric shapes.
- Density is the mass of an object divided by its volume.
- Population density can be determined by calculating the quotient of the number of people in an area and the area itself.
- Apply constraints to maximize or minimize the cost of a cardboard box used to package a product that represents a geometric figure. Apply volume relationships of cylinders, pyramids, cones, and spheres.
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TRANSLATE BETWEEN THE GEOMETRIC DESCRIPTION AND THE EQUATION FOR A CONIC SECTION |
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MGSE9-12.G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
KEY IDEAS
- A circle is the set of points in a plane equidistant from a given point, or center, of the circle.
- The standard form of the equation of a circle is (x – h)² + (y – k)² = r², where (h, k) is the center of the circle and r is the radius of the circle.
- The equation of a circle can be derived from the Pythagorean Theorem.
USE COORDINATES TO PROVE SIMPLE GEOMETRIC THEOREMS ALGEBRAICALLY |
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MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, sqrt{3}) lies on the circle centered at the origin and containing the point (0, 2).
MGSE9-12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
MGSE9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
MGSE9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
MGSE9-12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
MGSE9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
MGSE9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
KEY IDEAS
- Given the equation of a circle, you can verify whether a point lies on the circle by substituting the coordinates of the point into the equation. If the resulting equation is true, then the point lies on the figure. If the resulting equation is not true, then the point does not lie on the figure.
- Given the center and radius of a circle, you can verify whether a point lies on the circle by determining whether the distance between the given point and the center is equal to the radius.
- To prove properties about special parallelograms on a coordinate plane, you can use the partitioning of a segment, distance, and slope formulas.
• The partitioning of a segment formula is (x, y) = ( (bx1 + ax2)/(b + a) , (by1 + ay2)/(b + a)) or (x, y) = (x1 + (x2 + x1) a/(a + b), y1 + (y2 + y1) a/(a + b)). This formula is used to find the coordinates of a point which partitions a directed line segment AB at the ratio of a:b from A(x1, y1) to B(x2, y2). This formula can also be used to derive the midpoint formula.
• The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2) . This formula is used to find the coordinates of the midpoint of segment AB, given A(x1, y1) and B(x2, y2).
• The distance formula is d = sqrt{(x2 − x1)² + (y2 − y1)²}. This formula is used to find the length of segment AB, given A(x1, y1) and B(x2, y2).
• The slope formula is m = (y2 − y1)/(x2 − x1). This formula is used to find the slope of a line or line segment, given any two points on the line or line segment A(x1, y1) and B(x2, y2). Slopes can be positive, negative, 0, or undefined. - To prove a triangle is isosceles, you can use the distance formula to show that at least two sides are congruent.
- You can use properties of quadrilaterals to help prove theorems:
• To prove a quadrilateral is a parallelogram, show that the opposite sides are parallel using slope.
• To prove a quadrilateral is a rectangle, show that the opposite sides are parallel and the consecutive sides are perpendicular using slope.
• To prove a quadrilateral is a rhombus, show that all four sides are congruent using the distance formula.
• To prove a quadrilateral is a square, show that all four sides are congruent and consecutive sides are perpendicular using slope and the distance formula. - You can also use diagonals of a quadrilateral to help prove theorems:
• To prove a quadrilateral is a parallelogram, show that its diagonals bisect each other using the midpoint formula.
• To prove a quadrilateral is a rectangle, show that its diagonals bisect each other and are congruent using the midpoint and distance formulas.
• To prove a quadrilateral is a rhombus, show that its diagonals bisect each other and are perpendicular using the midpoint and slope formulas.
• To prove a quadrilateral is a square, show that its diagonals bisect each other, are congruent, and are perpendicular using the midpoint, distance, and slope formulas. - A directed line segment is a line segment from one point to another point in the coordinate plane.
- When using the formulas for partitioning of a segment, distance, and slope, the order of the points does not matter, but be careful to always subtract in the same order.
- Parallel lines have the same slope. Perpendicular lines have slopes that are the negative reciprocal of each other.
- When using directed line segments, pay close attention to the beginning and end points of the line. For example, the directed line segments PQ and QP have the same length but different directions.
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RESOURCES |
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complex_numbers_and_methods.pdf | |
File Size: | 225 kb |
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