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Question Number 203602 by mr W last updated on 22/Jan/24

Answered by mr W last updated on 23/Jan/24

Commented by mr W last updated on 23/Jan/24

$$makePE//CD$$$$\mathrm{\angle }EPD=\mathrm{\angle }PDC=60°=\mathrm{\angle }PDE$$$$\Rightarrow \mathrm{\Delta }PDEisequilateral.$$$$\Rightarrow PE=ED=PD$$$$\frac{BE}{BD}=\frac{EP}{DC}$$$$\frac{BD-ED}{BD}=\frac{EP}{DC}$$$$\frac{24-PD}{24}=\frac{PD}{8}$$$$4\times PD=24$$$$\Rightarrow PD=6✓$$

Answered by deleteduser1 last updated on 22/Jan/24

$$x=AB=BC=CA$$$$Ptolemy:8\times x+24\times x=x\left(DA\right)\Rightarrow DA=32$$$$Let\measuredangle CBD=\theta \Rightarrow \mathrm{\angle }BCD=60-\theta$$$${BC}^2={BD}^2+{DC}^2-2BD\times DCcos\left(120\right)$$$$\Rightarrow BC=8\sqrt{13}$$$$\frac{sin\left(\theta \right)}{8}=\frac{sin\left(120\right)}{8\sqrt{13}}\Rightarrow sin\left(\theta \right)=\frac{\sqrt{39}}{26}\Rightarrow cos\left(\theta \right)=\frac{7\sqrt{13}}{26}$$$$\frac{sin(120-\theta )}{24}=\frac{sin\left(\theta \right)}{DP}\Rightarrow DP=\frac{24sin\left(\theta \right)}{sin(120-\theta )}=6$$$$[sin(120-\theta )=\frac{8\sqrt{39}}{52}]$$

Commented by mr W last updated on 23/Jan/24

$$thanksalot!$$

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