Chapter 6, Problem 19E

To determine

To write:

A script that will prompt the user to enter the coordinates of three points that determine a triangle, then calculate and print the area of the triangle and then call one function to formulate the area of the triangle by calling a local function that calculates the side lengths formed by any two points.

Answer to Problem 19E

Solution:

The script file is,

% MATLAB code to calculate the area of the triangle by calling a function.

%script file.

x1 = input('the x coordinate of the fist point is entered:');

%enter the x coordinate of the first point.

x2 = input('the x coordinate of the second point is entered:');

%enter the x coordinate of the second point.

x3 = input('the x coordinate of the third point is entered:');

%enter the x coordinate of the third point.

y1 = input('the y coordinate of the fist point is entered:');

%enter the y coordinate of the first point.

y2 = input('the y coordinate of the second point is entered:');

%enter the y coordinate of the second point.

y3 = input('the y coordinate of the third point is entered:');

%enter the y coordinate of the third point.

Area = trianglearea(x1, y1, x2, y2, x3, y3);

fprintf('The triangle’’s area is %.2f\n', Area)

% end of script file

%The script file should be placed in the same folder.

The function file is,

% MATLAB code to calculate the area of the triangle.

%function file.

function out = trianglearea(x1, y1, x2, y2, x3, y3)

%define the area of triangle by using the function areatriangle.

a = sidetriangle(x1, y1, x2, y2);

%define the variable a.

b = sidetriangle(x2, y2, x3, y3);

%define the variable b.

c = sidetriangle(x3, y3, x1, y1);

%define the variable c.

s = (a+b+c)/2;

%define the variable s.

out = sqrt (s*(s-a)*(s-b)*(s-c));

end

function out2 = sidetriangle(x1, y1, x2, y2)

%calculate the distnace between the two points by calling a function

%sidetriangle.

out2 = sqrt((x1-x2)^2 + (y1-y2)^2);

%calculate the sides of the triangle.

end

% end of function

%The function file should be placed in the same folder.

Explanation of Solution

The given two points are $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$.

The formula of the distance between the two points is given as,

$\text{distance}=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$

Substitute 0 for $x_1$, 5 for $x_2$, 0 for $y_1$ and 0 for $y_2$ in the above formula.

$\begin{array}{l}\text{distance}=\sqrt{{\left(0-5\right)}^2+{\left(0-0\right)}^2}\\ \text{distance}=5\end{array}$

Consider the three points are $\left(0,0\right)$, $\left(5,0\right)$ and $\left(5,5\right)$.

The side of the triangle is,

$\text{a}=\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$

Substitute 0 for $x_1$, 5 for $x_2$, 0 for $y_1$ and 0 for $y_2$ in the above formula.

$\begin{array}{rll}\text{a}& =& \sqrt{{\left(0-5\right)}^2+{\left(0-0\right)}^2}\\ & =& 5\end{array}$

The side of the triangle is,

$\text{b}=\sqrt{{\left(x_2-x_3\right)}^2+{\left(y_2-y_3\right)}^2}$

Substitute 5 for $x_2$, 5 for $x_3$, 0 for $y_2$ and 5 for $y_3$ in the above formula.

$\begin{array}{rll}\text{b}& =& \sqrt{{\left(5-5\right)}^2+{\left(0-5\right)}^2}\\ & =& 5\end{array}$

The side of the triangle is,

$\text{c}=\sqrt{{\left(x_3-x_1\right)}^2+{\left(y_3-y_1\right)}^2}$

Substitute 5 for $x_3$, 0 for $x_1$, 5 for $y_3$ and 0 for $y_1$ in the above formula.

$\begin{array}{rll}\text{c}& =& \sqrt{{\left(5-0\right)}^2+{\left(5-0\right)}^2}\\ & =& 5\sqrt{2}\end{array}$

The formula for half sum of the sides of the triangle is,

$\text{s}=\frac{\text{a}+\text{b}+\text{c}}{2}$

Substitute 5 for a, 5 for b and $5\sqrt{2}$ for c in the above formula.

$\begin{array}{rll}\text{s}& =& \frac{5+5+5\sqrt{2}}{2}\\ & =& 8.5355\end{array}$

The formula for the area of the triangle is,

$\text{area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

Substitute 5 for a, 5 for b, $5\sqrt{2}$ for c and $8.5355$ for s in the above formula.

$\begin{array}{rll}\text{area}& =& \sqrt{8.5355\left(8.5355-5\right)\left(8.5355-5\right)\left(8.5355-5\sqrt{2}\right)}\\ & =& 12.5000\end{array}$

MATLAB Code:

% MATLAB code to calculate the area of the triangle by calling a function.

%script file.

x1 = input('the x coordinate of the fist point is entered:');

%enter the x coordinate of the first point.

x2 = input('the x coordinate of the second point is entered:');

%enter the x coordinate of the second point.

x3 = input('the x coordinate of the third point is entered:');

%enter the x coordinate of the third point.

y1 = input('the y coordinate of the fist point is entered:');

%enter the y coordinate of the first point.

y2 = input('the y coordinate of the second point is entered:');

%enter the y coordinate of the second point.

y3 = input('the y coordinate of the third point is entered:');

%enter the y coordinate of the third point.

Area = trianglearea(x1, y1, x2, y2, x3, y3);

fprintf('The triangle’’s area is %.2f\n', Area)

% end of script file

%The script file should be placed in the same folder.

% MATLAB code to calculate the area of the triangle.

%function file.

function out = trianglearea(x1, y1, x2, y2, x3, y3)

%define the area of triangle by using the function areatriangle.

a = sidetriangle(x1, y1, x2, y2);

%define the variable a.

b = sidetriangle(x2, y2, x3, y3);

%define the variable b.

c = sidetriangle(x3, y3, x1, y1);

%define the variable c.

s = (a+b+c)/2;

%define the variable s.

out = sqrt (s*(s-a)*(s-b)*(s-c));

end

function out2 = sidetriangle(x1, y1, x2, y2)

%calculate the distnace between the two points by calling a function

%sidetriangle.

out2 = sqrt((x1-x2)^2 + (y1-y2)^2);

%calculate the sides of the triangle.

end

% end of function

%The function file should be placed in the same folder.

Save the MATLAB script with name, mainscript.m, and the function file is trianglearea.m in the current folder. Execute the program by typing the script name at the command window to generate result.

Result:

The output is,

Therefore, the result is stated above.