Solve {l}{a=10,b=15,c=21}{p=1/2(a+b+c)}{S=sqrt{(p-a)(p-b)(p-c)}}

$$\left. \begin{array} { l } { a = 10 , b = 15 , c = 21 } \\ { p = \frac { 1 } { 2 } ( a + b + c ) } \\ { S = \sqrt { ( p - a ) ( p - b ) ( p - c ) } } \end{array} \right.$$

Solve for a, b, c, p, S

$$a=10$$

$$b=15$$

$$c=21$$

$$p=23$$

$$S=4\sqrt{13}\approx 14.422205102$$

Solution Steps

Consider the fourth equation. Insert the known values of variables into the equation.

$$p=\frac{1}{2}\left(10+15+21\right)$$

Add 10 and 15 to get 25.

$$p=\frac{1}{2}\left(25+21\right)$$

Add 25 and 21 to get 46.

$$p=\frac{1}{2}\times 46$$

Multiply \frac{1}{2} and 46 to get 23.

$$p=23$$

Consider the fifth equation. Insert the known values of variables into the equation.

$$S=\sqrt{\left(23-10\right)\left(23-15\right)\left(23-21\right)}$$

Subtract 10 from 23 to get 13.

$$S=\sqrt{13\left(23-15\right)\left(23-21\right)}$$

Subtract 15 from 23 to get 8.

$$S=\sqrt{13\times 8\left(23-21\right)}$$

Multiply 13 and 8 to get 104.

$$S=\sqrt{104\left(23-21\right)}$$

Subtract 21 from 23 to get 2.

$$S=\sqrt{104\times 2}$$

Multiply 104 and 2 to get 208.

$$S=\sqrt{208}$$

Factor 208=4^{2}\times 13. Rewrite the square root of the product \sqrt{4^{2}\times 13} as the product of square roots \sqrt{4^{2}}\sqrt{13}. Take the square root of 4^{2}.

$$S=4\sqrt{13}$$

The system is now solved.

$$a=10 b=15 c=21 p=23 S=4\sqrt{13}$$