Question

Find the output of the program given below. 10 REM Area of a triangle when the sides are given \(20 \mathrm{READ} \times, \mathrm{y}, \mathrm{z}\) 30 DATA \(8,15,17\) 40 Let \(S=(x+y+z) / 2\) 50 Let \(\mathrm{A}=\left[\mathrm{S}^{\star}(\mathrm{S}-\mathrm{x}) \star(\mathrm{S}-\mathrm{y}) \star(\mathrm{S}-\mathrm{Z})\right]^{1 / 2}\) 60 PRINT "The area \(=", A\) 70 END (1) 120 (2) 60 (3) 90 (4) The triangle is not feasible

Short answer

Answer: (2) 60

Step-by-step solution

Read and assign values

The program reads the values x=8, y=15, and z=17 from the DATA statement in line 30.

Calculate the semi-perimeter

According to the program, in line 40, the semi-perimeter $$S$$ is calculated as follows: $$S = (x+y+z)/2$$, so put the values of x, y, and z from step 1: $$S = (8+15+17) / 2 = 20$$

Calculate the area

In line 50, the program computes the area A using Heron's formula: $$A = [S(S-x)(S-y)(S-z)]^{1/2}$$ Plug in the values obtained in the previous steps: $$A = [20(20-8)(20-15)(20-17)]^{1/2} = [20(12)(5)(3)]^{1/2} = 60$$

Match the output with the given options

According to the calculated area in step 3, the area A is equal to 60. Comparing this result with the given options, we can conclude that the output of the program is option (2) 60.

Key concepts

Area of a triangle

Calculating the area of a triangle can be slightly tricky when only the side lengths are known. In such cases, Heron's formula is incredibly useful. This formula allows us to find the area without needing any angle measurements. Here's how it works: once you know the lengths of all three sides of a triangle, you can apply Heron's formula.
The formula is given by: \[ A = \sqrt{S(S-x)(S-y)(S-z)} \] where \( A \) is the area of the triangle, \( S \) is the semi-perimeter of the triangle, and \( x, y, z \) are the side lengths.

• This formula can be applied to any type of triangle, which makes it extremely versatile.
• The key to using it correctly is ensuring precise calculations, especially since it involves taking a square root.

Using Heron's formula, we can calculate the area for our triangle with sides 8, 15, and 17 as shown in the solution, resulting in an area of 60 square units.

Programming in BASIC

BASIC is a beginner-friendly programming language, ideal for learning computational logic. The language's simplicity makes it an excellent tool for implementing mathematical computations like finding the area of a triangle.
The program provided in the exercise uses BASIC to read values, compute the semi-perimeter, and then calculate the area using Heron's formula. BASIC's straightforward syntax supports learning programming fundamentals.

• Commands in BASIC are usually easy to understand, like 'READ' for input and 'PRINT' for output.
• Line numbers structure the code, which allows for clear and sequential execution.
• The 'LET' command is used as an assignment operator to define variables.

With its simple structure, BASIC provides a solid starting point for learning how to systematically translate mathematical methods into programming procedures.

Semi-perimeter calculation

The semi-perimeter is a critical intermediary value in Heron's formula. It's essentially half of the triangle's perimeter. Calculating it is a crucial first step when using Heron's formula to find the area.
To calculate the semi-perimeter, simply add all three side lengths together and divide by two: \[ S = \frac{(x + y + z)}{2} \] where \( S \) is the semi-perimeter and \( x, y, z \) are the triangle's sides.

• The semi-perimeter acts as a key component in Heron's formula for determining the area.
• It simplifies the multiplication of terms within the square root, making the formula more efficient.

In our example, with side lengths 8, 15, and 17, the semi-perimeter is calculated to be 20. This calculation enables the next step of determining the area.