Pentagon

The signboard has the shape of a pentagon ABCDE, in which line BC is perpendicular to line AB, and EA is perpendicular to line AB. Point P is the heel of the vertical starting from point D on line AB. | AP | = | PB |, | BC | = | EA | = 6dm, | PD | = 8.4dm. Point X - the intersection of the lines PE and DA, the point Y - the intersection of the lines PC and BD is marked on the shield. I need to focus the point Z, which is the intersection of the lines XY and PD. Specify | PZ |.

Correct answer:

z =  3.5 dm

Step-by-step explanation:

$$\begin{gathered} a=6\text{dm} \\ b=8.4\text{dm} \\ a:(A+B)=z:B \\ b:(A+B)=z:A \\ a\cdot B=z\cdot A+z\cdot B \\ b\cdot A=z\cdot A+z\cdot B \\ a\cdot B=b\cdot A \\ A:B=a:b \\ a:(A+B)=z:B \\ z=a\cdot \frac{B}{A+B} \\ z=a\cdot \frac{1}{A/B+1} \\ z=a\cdot \frac{1}{a/b+1}=6\cdot \frac{1}{6/8.4+1}=\frac{7}{2}=3.5=\frac{7}{2}\text{dm}=3.5\text{dm} \\ \text{ Verifying Solution: } \\ {z}_2=b\cdot \frac{1}{1+b/a}=8.4\cdot \frac{1}{1+8.4/6}=\frac{7}{2}=3.5\text{dm} \end{gathered}$$

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