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This system of units is based on the yard for length and the pound for weight. The main units of length are inch, foot, yard, and mile. The relationships between the units
of length are as follows:
12 inches = 1 foot 3 feet = 1 yard 22 yards = 1 chain 10 chains = 1 furlong 22 furlongs = 1 mile 1760 yards = 1 mile
The main units of weight are ounce (oz), pound (lb), hundredweight (cwt), and ton:
16 oz = 1 lb 112 lb = 1 cwt 20 cwt = 1 ton 2240 lb = 1 ton
The units for capacity of liquids are pint, quart, and gallon:
2 pints = 1 quart 4 quarts = 1 gallon 8 pints = 1 gallon
For area, there are 4840 square yards in 1 acre.
Reference: Metric Units.
The incenter is the center of the inscribed circle of a triangle, which is also known as the incircle of a triangle. In the figure the point is the incenter of the triangle. The inscribed circle of a triangle is a circle that lies inside a triangle and touches each of the three sides of the triangle. The radius of the incircle is called the inradius. Each triangle has only one incircle and therefore only one incenter. The three sides of the triangle are tangents to its incircle.
INCONSISTENT EQUATIONS 247
On the left-hand side of the figure the incenter of the triangle is found by constructing the angle bisectors (drawn dashed in the figure) of each of the angles of the triangle. In practice, only two angle bisectors will be needed to find the incenter of a triangle, because the third angle bisector will also pass through the incenter. On the right-hand side of the figure the radius of the incircle is R, and the three radii drawn dashed are at right angles to the sides of the triangle.
References: Angle Bisector, Radius, Tangent.
This is a plane that is not horizontal or vertical. In the figure the roof of the house is an inclined plane, because it is sloping. The wall of the house is a vertical plane and the floor of the house is a horizontal plane.
Two equations are inconsistent if they cannot both be true together. We also can say that the two equations are incompatible. For example, the two equations
2x + 3y = 5 and 2x + 3y = 6
are inconsistent equations, because they both cannot be true together: 2x + 3 y cannot be equal to two different quantities at the same time. For example, suppose Nathan
248 INDIRECT TRANSFORMATION
and Jacob went to the cafe for lunch. Nathan had two cakes and three buns and his bill was $5. Jacob had exactly the same food, two cakes and three buns, and his bill was $6. If the buns and cakes were the same type for each person, then the results are inconsistent.
Reference: Decreasing Function.
Reference: Complementary Events.
Reference: Dependent Variable.
The plural of index is indices.
Under the entry Congruent Figures, two kinds of congruences are explained:
Directly congruent direct transformations
Oppositely congruent indirect transformations
For a direct transformation the object and the image are directly congruent, and for an indirect transformation the object and the image are oppositely congruent. Reflection is an indirect transformation and direct transformations are rotation, translation, and enlargement.
Another way of recognizing when a transformation is direct or indirect is to check the object and the image in the following way: In the figure the triangle ABC is reflected in the mirror line m and its image is triangle A’B'C'. If you move your finger around the triangle ABC in the alphabetical order of the letters you will move in a counterclockwise direction. But if you move your finger around the image triangle A'B'C' in alphabetical order you will move in a clockwise direction. This is a property of all indirect transformations. For all direct transformations the alphabetical direction described here is not reversed.
If you place together a pair of your shoes, one of them is an indirect transformation of the other. In this example the transformation is a reflection. If you moved only one shoe to another place, the transformation could never be a single reflection.
References: Congruent Figures, Enlargement, Image, Object, Rotation, Transformation Geometry, Translation.
Under this entry we study the graphs of the following inequalities:
♦ Greater than (>)
♦ Greater than or equal to (>)
♦ Less than (<)
♦ Less than or equal to (<)
We will graph four inequations, and then show how these four inequations can be used to solve a problem. First be certain you know how to draw the graphs of straight lines. Refer to the Gradient-Intercept Form of a straight line.
Example 1. Sketch the following inequations on separate axes:
(a) x > 3.