Question
Using the following figure, show that $BD =\sqrt{x}$.
✓
Answer
$ \mathrm{AB}=\mathrm{x}$ and $\mathrm{BC}=1$
$ \mathrm{AC}=\mathrm{AB}+\mathrm{BC}$
$ =\mathrm{x}+1$
diameter$=\mathrm{x}+1$
radius $\mathrm{OA}=\mathrm{OD}=\mathrm{OC}=\mathrm{OB}=\mathrm{OC}-\mathrm{BC}$
$ =\frac{x+1}{2}-1=\frac{x+1-2}{2}=\frac{x-1}{2}$
Using pythagoras in $\triangle \mathrm{BOD}$
$\mathrm{P}^2+\mathrm{B}^2=\mathrm{H}^2$
$ \mathrm{P}^2+\left(\frac{x-1}{2}\right)^2=\left(\frac{x+1}{2}\right)^2$
$ \mathrm{P}^2=\left(\frac{x+1}{2}\right)^2-\left(\frac{x-1}{2}\right)^2$
$ =\frac{(x+1)^2-(x-1)^2}{4}$
$ =\frac{\left(x^2+1+2 x\right)-\left(x^2+1-2 x\right)}{4}$
$\mathrm{P}^2=\frac{4 x}{4}$
$P^2=x$
$\mathrm{P}=\sqrt{x}$
$ \mathrm{AC}=\mathrm{AB}+\mathrm{BC}$
$ =\mathrm{x}+1$
diameter$=\mathrm{x}+1$
radius $\mathrm{OA}=\mathrm{OD}=\mathrm{OC}=\mathrm{OB}=\mathrm{OC}-\mathrm{BC}$
$ =\frac{x+1}{2}-1=\frac{x+1-2}{2}=\frac{x-1}{2}$
Using pythagoras in $\triangle \mathrm{BOD}$
$\mathrm{P}^2+\mathrm{B}^2=\mathrm{H}^2$
$ \mathrm{P}^2+\left(\frac{x-1}{2}\right)^2=\left(\frac{x+1}{2}\right)^2$
$ \mathrm{P}^2=\left(\frac{x+1}{2}\right)^2-\left(\frac{x-1}{2}\right)^2$
$ =\frac{(x+1)^2-(x-1)^2}{4}$
$ =\frac{\left(x^2+1+2 x\right)-\left(x^2+1-2 x\right)}{4}$
$\mathrm{P}^2=\frac{4 x}{4}$
$P^2=x$
$\mathrm{P}=\sqrt{x}$
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Case Study I: The union of the set of rational numbers (Q) and irrational numbers (P) form the set of real numbers (R). Rational numbers are the set of numbers which can be written in the form $\frac{a}{b}$ , where a and b are integers and $b \neq 0$. The decimal expansion of a rational number is either terminating or non-terminating repeating. The number which cannot be expressed in the form $\frac{a}{b}$ are called irrational numbers. The decimal expansion of irrational numbers is non-terminating non-repeating.
Based on the above information, answer the following questions:
1. Every rational number is:
(a) a natural number (b) a whole number
(c) an integer (d) a real number
2. Every real number is:
(a) an integer
(b) a rational number
(c) an irrational number
(d) either a rational number or an irrational number.
3. The sum of two irrationals is:
(a) irrational (b) rational
(c) either rational or irrational (d) neither rational nor irrational
4. The product of a rational and an irrational number is:
(a) an irrational number
(b) a rational number
(c) either a rational number or an irrational number
(d) neither a rational number nor an irrational number
5. The number of irrational numbers is:
(a) finite (b) infinite
(c) neither finite nor infinite (d) none of these