Question-215454

Question Number 215454 by cherokeesay last updated on 07/Jan/25

Answered by mr W last updated on 07/Jan/25

Commented by mr W last updated on 07/Jan/25

EB=2a EA=2b CB=(√3)a DA=(√3)b (1/2)×(2b−a)(√3)a=2×8 (1/2)×(2a−b)(√3)b=2×15 (((2a−b)b)/((2b−a)a))=((15)/8) 15((a/b))^2 −14((a/b))−8=0 ⇒(a/b)=((7+(√(49+15×8)))/(15))=(4/3) (1/2)×(2×((4b)/3)−b)(√3)b=2×15 (√3)b^2 =36 ⇒b=(6/( (√(√3)))) ⇒a=(8/( (√(√3)))) ΔABE=2×15+((b^2 (√3))/2)=30+18=48 ?=48−8−15=25 ✓

$${EB}=\mathrm{2}{a} \\ $$$${EA}=\mathrm{2}{b} \\ $$$${CB}=\sqrt{\mathrm{3}}{a} \\ $$$${DA}=\sqrt{\mathrm{3}}{b} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\left(\mathrm{2}{b}−{a}\right)\sqrt{\mathrm{3}}{a}=\mathrm{2}×\mathrm{8} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\left(\mathrm{2}{a}−{b}\right)\sqrt{\mathrm{3}}{b}=\mathrm{2}×\mathrm{15} \\ $$$$\frac{\left(\mathrm{2}{a}−{b}\right){b}}{\left(\mathrm{2}{b}−{a}\right){a}}=\frac{\mathrm{15}}{\mathrm{8}} \\ $$$$\mathrm{15}\left(\frac{{a}}{{b}}\right)^{\mathrm{2}} −\mathrm{14}\left(\frac{{a}}{{b}}\right)−\mathrm{8}=\mathrm{0} \\ $$$$\Rightarrow\frac{{a}}{{b}}=\frac{\mathrm{7}+\sqrt{\mathrm{49}+\mathrm{15}×\mathrm{8}}}{\mathrm{15}}=\frac{\mathrm{4}}{\mathrm{3}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\left(\mathrm{2}×\frac{\mathrm{4}{b}}{\mathrm{3}}−{b}\right)\sqrt{\mathrm{3}}{b}=\mathrm{2}×\mathrm{15} \\ $$$$\sqrt{\mathrm{3}}{b}^{\mathrm{2}} =\mathrm{36} \\ $$$$\Rightarrow{b}=\frac{\mathrm{6}}{\:\sqrt{\sqrt{\mathrm{3}}}}\:\Rightarrow{a}=\frac{\mathrm{8}}{\:\sqrt{\sqrt{\mathrm{3}}}} \\ $$$$\Delta{ABE}=\mathrm{2}×\mathrm{15}+\frac{{b}^{\mathrm{2}} \sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{30}+\mathrm{18}=\mathrm{48} \\ $$$$?=\mathrm{48}−\mathrm{8}−\mathrm{15}=\mathrm{25}\:\checkmark \\ $$

Commented by mr W last updated on 07/Jan/25