91影视

Suppose a triangle has side lengths a, b and c. The semi-perimeter of the triangle is defined to be $s=\frac{1}{2}(a+b+c)$. We can use the side lengths of the triangle to calculate its area by applying Heron鈥檚 formula:

$\mathrm{Area}=\sqrt{s(s鈭�a)(s鈭�b)(s鈭�c)}$

Use Heron鈥檚 formula to compute the areas of the triangles determined by the points P, Q and R in Exercises 42鈥�44:

P, Q and R from Exercise 37

The herons formula$\mathrm{Area}=\sqrt{s(s鈭�a)(s鈭�b)(s鈭�c)}$

Where $s=\frac{1}{2}(a+b+c)$

$$\begin{gathered} a=\mathrm{Distance}\mathrm{between}QR \\ b=\mathrm{Distance}\mathrm{between}PR \\ c=\mathrm{Distance}\mathrm{between}PQ \end{gathered}$$

$$\begin{gathered} a=QR=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2+{(z_2-z_1)}^2} \\ a=QR=\sqrt{{\left\{\right(-1)-0\}}^2+{(3-0)}^2+{(6-3)}^2} \\ a=QR=\sqrt{{(-1)}^2+{\left(3\right)}^2+{\left(3\right)}^2} \\ a=QR=\sqrt{1+9+9} \\ a=QR=\sqrt{19} \\ a=QR鈮�4.36 \end{gathered}$$

$$\begin{gathered} b=PR=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2+{(z_2-z_1)}^2} \\ b=PR=\sqrt{{\left\{\right(-1)-4\}}^2+{(3-3)}^2+{\{6-(-2\left)\right\}}^2} \\ b=PR=\sqrt{{(-5)}^2+{\left(0\right)}^2+{\left(8\right)}^2} \\ b=PR=\sqrt{25+0+64} \\ b=PR=\sqrt{89} \\ b=PR鈮�9.43 \end{gathered}$$

$$\begin{gathered} c=PQ=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2+{(z_2-z_1)}^2} \\ c=PQ=\sqrt{{(0-4\}}^2+{(0-3)}^2+{\{3-(-2\left)\right\}}^2} \\ c=PQ=\sqrt{{(-4)}^2+{(-3)}^2+{(-5)}^2} \\ c=PQ=\sqrt{16+9+25} \\ c=PQ=\sqrt{50} \\ c=PQ鈮�7.07 \end{gathered}$$

$$\begin{gathered} s=\frac{a+b+c}{2} \\ s=\frac{4.36+9.43+7.07}{2} \\ s=\frac{20.86}{2} \\ s=10.43 \end{gathered}$$

$$\begin{gathered} \mathrm{Area}=\sqrt{s(s-a)(s-b)(s-c)} \\ \mathrm{Area}=\sqrt{10.43(10.43-4.36)(10.43-9.43)(10.43-7.07)} \\ \mathrm{Area}=\sqrt{10.43脳6.07脳1脳3.36} \\ \mathrm{Area}=\sqrt{212.72} \\ \mathrm{Area}鈮�14.58 \end{gathered}$$