# G.3a and *G.3a - Find Distance, Midpoint

G.3    The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include

a)  investigating and using formulas for finding distance, midpoint, and slope;

G.3  The student will solve problems involving symmetry and transformation. This will include

a)  investigating and using formulas for determining distance, midpoint, and slope;

Bloom's Level:  Apply

### BIG IDEAS

• I can master the challenge of a ski slope for snowboarding, determine the gradient of a road, figure the pitch of a roof, and build a handicap accessible ramp to a door.  I can design a wallpaper pattern, tile a floor, create a quilt, and piece together a stain glass window.
• I will apply slope as rate of change where one value changing proportionately effects the other value, and the pattern of this relationship can be represented by a line that facilitates analysis and prediction.  I will understand that an ordered pair represents a unique location in space and be able to find specific locations on a grid and see how moving an object does not change its size or shape.

### UNDERSTANDING THE STANDARD

• Transformations and combinations of transformations can be used to describe movement of objects in a plane.
• The distance formula is an application of the Pythagorean Theorem.
• Geometric figures can be represented in the coordinate plane.
• Techniques for investigating symmetry may include paper folding, coordinate methods, and dynamic geometry software.
• Parallel lines have the same slope.
• The product of the slopes of perpendicular lines is -1.
• The image of an object or function graph after an isomorphic transformation is congruent to the preimage of the object.

·  Symmetry and transformations can be explored with computer software, paper folding, and coordinate methods.

·  The distance formula is an application of the Pythagorean Theorem.

·  Geometric figures can be represented in the coordinate plane.

·  Parallel lines have the same slope.

·  The product of the slopes of perpendicular lines is −1 unless one of the lines has an undefined slope.

·  A transformation of a figure, called a preimage, changes the size, shape, and/or position of the figure to a new figure called the image.

·  Transformations and combinations of transformations can be used to describe movement of objects in a plane.

·  The image of an object or function graph after an isomorphic transformation is congruent to the preimage of the object.

­  A rotation is an isomorphic transformation in which an image is formed by rotating the preimage about a point called the center of rotation. The center of rotation may or may not be on the preimage.  Rotations may be more than 180⁰.

­  A reflection is an isomorphic transformation in which an image is formed by reflecting the preimage over a line called the line of reflection. All corresponding points in the image are equidistant from the line of reflection.

­   A translation is an isomorphic transformation in which an image is formed by moving every point on the preimage the same distance in the same direction.

·  A dilation is a transformation in which an image is formed by enlarging or reducing the preimage proportionally by a scale factor from the center of dilation. The center of dilation may or may not be on the preimage. The image is similar to the preimage.

### ESSENTIALS

G.3a3  Apply the distance formula to find the length of a line segment when given the coordinates of the endpoints. (Show using the Pythagorean Theorem)

·  G.3a3  Apply the distance formula to determine the length of a line segment when given the coordinates of the endpoints.

G.3a1  Find the coordinates of the midpoint of a segment, using the midpoint formula

·  G.3a1  Determine the coordinates of the midpoint or endpoint of a segment, using the midpoint formula.

### KEY VOCABULARY

pictorial representation, computer software, constructions, coordinate method, number line, symmetry, transformation, distance, midpoint, slope, distance formula, midpoint formula, slope formula, line, parallel, perpendicular, line symmetry, point symmetry, point, translation, reflection, rotation, dilation, image, pre-image, coordinates, endpoints, combination, plane, Pythagorean Theorem, line segment, geometric figure, coordinate plane, product, function, isomorphic/isometry, congruent

Updated: Oct 27, 2017