GOING THE DISTANCE

GOALS

1. read and interpret maps

2. understand different ways in which distance is represented and used

3. use the relationship between the side length and the area of a square

4. understand and use the Pythagorean theorem

5. use formulas to solve problems

6. formalize vector use, including how to find the resultant vector

7. develop procedures for finding the areas of triangles, polygons, and circles

8. use vectors to solve problems

9. understand and use constructions to solve distance problems

10. choose appropriate models or views to solve three-dimensional distance or steepness problems

11. understand slope, tangent, and glide angles and use relationships among them to solve problems

SECTION SUMMARIES

SECTION A: EQUAL DISTANCES

1. Points equidistant from two points form a straight line; this line is the perpendicular bisector of the segment

joining the two points.

2. In a plane, all points equidistant from one point form a circle.

SECTION B: FINDING YOUR WAY

1. A vector is composed of two parts: a direction, usually in degrees, and a distance, usuall in either kilometers

or miles.

2. An arrow is used to represent a vector. Vector notation shows heading and distance:

for example, 240°/28 km.

3. A resultant vector describes the direction and distance from the start of the first vector to the end of the last vector.

SECTION C: DISTANCES: PYTHAGORAS

1. The Pythagorean theorem can be used to find the length of any one side of a right triangle if the lengths of the other two sides are known.

2. The Pythagorean theorem can be written as the equation a + b = c , where a and b represent the two smallest sides, called legs, and c represents the longest side, called the hypotenuse (the hypotenuse is always opposite the right angle).

3. If the Pythagorean theorem is not true for a given triangle, then that triangle is not a right triangle.

SECTION D: GLIDING THE DISTANCE

1. Glide ratio, slope, and tangent are all different expressions describing the same concept: the ratio height to

horizontal distance.

2. The larger the ratio, the larger the angle. We can use this relationship to find the measure of the angle if we know the side lengths or we can find a missing side length if we know the ratio and the other side length.

SECTION E: SLOPES AND DISTANCES

1. Contour lines on a map connect points of equal altitude. The numbers show altitudes or heights, reltive to sea level, at given intervals. When the contour lines are close, the land is very steep; when the lines are spread apart, the land rises more gently.

SECTION F: DISTANCES AND AREAS

1. In every triangle, you can draw a line from each vertex perpendicular to the opposite side. This opposite side

is referred to as the base. The length of the perpendicular line is called the height, or altitude, of a triangle.

2. Three formulas you’ve studied: Triangle Area = ½ bh Circumference of a circle = 2πr or πd

Area of a circle = πr