91影视

Using the results of Exercise 45 we have to find the areas of the quadrilaterals $PQRS$

$PQ=6,QR=7,RS=8,SP=9,QS=10$

To find the area of quadrilateral we firstly finding the area of $鈭�QPS\mathrm{and}鈭�QRS$then add them.

The area of$鈭�QPS=\sqrt{s(s-a)(s-b)(s-c)}$

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

From the diagramrole="math" localid="1649739691369" $a=6,b=9,c=10$

Sorole="math" localid="1649739746415" $s=\frac{6+9+10}{2}=\frac{25}{2}$

$$\begin{gathered} \mathrm{Area}鈭�QPS=\sqrt{s(s-a)(s-b)(s-c)} \\ \mathrm{Area}鈭�QPS=\sqrt{\frac{25}{2}\left(\frac{25}{2}-6\right)\left(\frac{25}{2}-9\right)\left(\frac{25}{2}-10\right)} \\ \mathrm{Area}鈭�QPS=\sqrt{\frac{25}{2}\left(\frac{25-12}{2}\right)\left(\frac{25-18}{2}\right)\left(\frac{25-20}{2}\right)} \\ \mathrm{Area}鈭�QPS=\sqrt{\frac{25}{2}脳\frac{13}{2}脳\frac{7}{2}脳\frac{5}{2}} \\ \mathrm{Area}鈭�QPS=\frac{5}{2脳2}\sqrt{13脳7脳5} \\ \mathrm{Area}鈭�QPS=\frac{5}{4}\sqrt{455} \end{gathered}$$

From the diagram$a=7,b=8,c=10$

So

$$\begin{gathered} s=\frac{7+8+10}{2} \\ s=\frac{25}{2} \end{gathered}$$

$$\begin{gathered} \mathrm{Area}鈭�QPS=\sqrt{s(s-a)(s-b)(s-c)} \\ \mathrm{Area}鈭�QPS=\sqrt{\frac{25}{2}\left(\frac{25}{2}-7\right)\left(\frac{25}{2}-8\right)\left(\frac{25}{2}-10\right)} \\ \mathrm{Area}鈭�QPS=\sqrt{\frac{25}{2}\left(\frac{25-14}{2}\right)\left(\frac{25-16}{2}\right)\left(\frac{25-20}{2}\right)} \\ \mathrm{Area}鈭�QPS=\sqrt{\frac{25}{2}脳\frac{11}{2}脳\frac{9}{2}脳\frac{5}{2}} \\ \mathrm{Area}鈭�QPS=\frac{5脳3}{2脳2}\sqrt{55} \\ \mathrm{Area}鈭�QPS=\frac{15}{4}\sqrt{55} \end{gathered}$$

$$\begin{gathered} \mathrm{Area}\mathrm{of}PQRS=\mathrm{Area}\mathrm{of}鈭�QPS+\mathrm{Area}\mathrm{of}鈭�QPS \\ \mathrm{Area}\mathrm{of}PQRS=\frac{5}{4}\sqrt{455}+\frac{15}{4}\sqrt{55} \\ \mathrm{Area}\mathrm{of}PQRS=\frac{5}{4}\left(\sqrt{455}+3\sqrt{55}\right) \end{gathered}$$