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Text Box Provides a means of entering and displaying text. The text can be assigned initially, it can be reassigned during program execution, or it can be entered by the user during program execution. (See also the Label Box and the Combo Box descriptions.) Timer Allows events to occur repeatedly at specified time intervals. Vertical Scroll Bar Allows a vertical scroll bar to be added to a control (if a vertical scroll bar is not included automatically).

To Prove: (d) In Fig. 3-35: Given:

a with 5 elements: a [ 0 ] contains 11.11, a [ l] contains 33.33, a [ 2 ] contains 55.55, a [ 3 ] contains 77.77, and a [ 4 ] contains 99.99. The diagram actually represents a region of the computer s memory because an array is always stored this way with its elements in a contiguous sequence.

Fig. 3-34

The method of numbering the ith element with index i-l is called zero-based indexing. Its use has the effect that the index of an array element is always the same as the number of steps from the initial element a [ o ] to that element. For example, element a [ 3 ] is 3 steps from element a [ o ] . The advantage of this method will become more apparent in 6 when we see the relationship between arrays and pointers.

Fig. 3-35

Virtually all useful programs use arrays. One reason that arrays are so useful is that they allow a single name with a variable index to be used in place of many different names. This makes it easy to do many things that would be far more difficult without arrays.

Prove each of the following: (a) The median to the base of an isosceles triangle bisects the vertex angle.

This program reads 4 numbers and then prints them in reverse order:

main0 1 double a[4]; tout cc "Enter 4 real numbers:\n'; for (int i = 1; i C= 4; i++) { tout CC i CC ": '; tin >> a[i-11; > tout cc "Here they are in reverse order:\n"; for (i = 3; i >= 0; i--) tout CC "\ta[" CC i CC "1 = ' CC a[i] -KC endl;

(3.12)

The declaration double a [ 4 ] declares a to be an array of 4 elements of type double. The first for loop then allows the user to enter real numbers into these 4 components. Then the second for loop prints these stored numbers in reverse order.

CHAP. 4]

(b) If the bisector of an angle of a triangle is also an altitude to the opposite side, then the other two sides of the triangle are congruent. (c) If a median to a side of a triangle is also an altitude to that side, then the triangle is isosceles. (d) In an isosceles triangle, the medians to the legs are congruent. (e) In an isosceles triangle, the bisectors of the base angles are congruent.

The array looks like this:

Parallel lines are straight lines which lie in the same plane and do not intersect however far they are ex4 4 4 4 tended. The symbol for parallel is i; thus, AB iCD is read line AB is parallel to line CD. In diagrams, arrows are used to indicate that lines are parallel (see Fig. 4-1).

a 1.618 2.718

Fig. 4-1

The next example works the same way. But it uses a symbolic constant for the size of the array. This makes the code easier to modify.

main0 1 const int size = 4; double a[size]; tout << "Enter ' C-C size XC ' real numbers:\n"; for (int i = 1; i <= size; i++) { co& cc i -c-c 1': It; tin >> a[i-11; 1 tout << "Here they are in reverse order:\n"; for (i = size-l; i >= 0; i--) tout << "\ta[U << i << "1 = I' << a[i] -C-C endl; 1

Fig. 4-2

The constant integer s i ze is initialized with the value 4. It is then used to declare the array a, to prompt the user, and to control both for loops. The program works the same as the previous version.