But you can be more exact than this. In Fig 1. To prove this, see Miscellaneous exercise 1 Question Notice that the condition does not work if the lines are parallel to the axes. You could tackle this question in several ways. This solution shows that the points form a parallelogram, and then that its diagonals are perpendicular. Therefore the parallelogram is a rhombus. Always draw a diagram, like Fig. First find the gradient of BC and its equation.
Exercise 1C 1 In each part write down the gradient of a line which is perpendicular to one with the given gradient. Find also the point of intersection of the two lines. Find its area. Find the coordinates of its vertices.
Find the coordinates of the mid-point of AC. Use your answer to find the coordinates of D. Calculate the coordinates of D. Calculate the coordinates of B and C.
Published by Hodder Education. In Fig. Show that the triangle TMN is isosceles. Ian Stewart. Be the first to like this. The point 2,3 is a vertex of the parallelogram. Rating details.
Calculate the area of the triangle ABC. Show that the medians are concurrent all pass through the same point. Prove that the lines are perpendicular. Chapter 2 Surds and indices The first part of this chapter is about expressions involving square and cube roots. These are natural numbers, or positive integers.
Then it was found that numbers could also be useful for measurement and in commerce. For these purposes fractions were also needed. Integers and fractions together make up the rational numbers. These are numbers which can be expressed p in the form q where p and q are integers, and q is not 0. One of the most remarkable discoveries of the ancient Greek mathematicians was that there are numbers which cannot be expressed in this way. These are called irrational numbers. The argument that the Greeks used to prove that 2 cannot be expressed as a fraction can be adapted to show that the square root, cube root, … of any positive integer is either an integer or an irrational number.
Rational and irrational numbers together make up the real numbers. Integers, rational and irrational numbers, and real numbers can be either positive, negative or zero.
When rational numbers are written as decimals, they either come to a stop after a number of places, or the sequence of decimal digits eventually starts repeating in a regular pattern. For example, The reverse is also true: if a decimal number stops or repeats indefinitely then it is a rational number. So if an irrational number is written as a decimal, the pattern of the decimal digits never repeats however long you continue the calculation.
You might have written 2. Expressions like 2 or 3 9 are called surds. This section is about calculating with surds. The main properties of surds that you will use are:. The height of the roof, BD, is 10 m. Calculate x and y. Giving each answer in simplified surd form, find a the area of the rectangle,. The examples in Questions 15 and 16 indicate a method for rationalising the denominator in cases which are more complicated than those in Question 5. They found it was more economical to write and to print the products xxx and xxxx as x 3 and x 4.
This is how index notation started. But it turned out to be much more than a convenient shorthand. The new notation led to important mathematical discoveries, and mathematics as it is today would be inconceivable without index notation. You will already have used simple examples of this notation. Notice that, although a can be any kind of number, m must be a positive integer.
Expressions in index notation can often be simplified by using a few simple rules. In explaining these rules multiplication signs have been used. But, as in other parts of algebra, they are usually omitted if there is no ambiguity. For completeness, here are the rules again.
But extending the meaning of am when the index is zero or negative is possible, and useful, since it turns out that the rules still work with such index values. On the left sides, the base is always 2, and the indices go down by 1 at each step. On the right, the numbers are halved at each step.
So you might continue the process … , Here are some examples to show that, with these definitions, the rules established in Section 2. Try making up other examples for yourself.
One application of negative indices is in writing down very small numbers. You probably know how to write very large numbers in standard form, or scientific notation. For example, it is easier to write the speed of light as 3. Similarly, the wavelength of red light, about 0. Computers and calculators often give users the option to work in scientific notation, and if numbers become too large or too small to be displayed in ordinary numerical form they will switch into standard form.
Some older calculators display, for example 3.
Find the radius of the cross-section. What happens if m and n are not necessarily integers? A slight extension of the x n n x rule can show you how to deal with expressions of 2 the form x 3. There are two alternatives: 2. If x has an exact cube root it is usually best to use the first form; otherwise the second form is better. In general, similar reasoning leads to the fractional power rule. There are often good alternative ways for solving problems involving indices, and you should try experimenting with them. Stimulating worked examples take a step-by-step approach to problem solving.
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