Question

in the figure ABllPQllCD,prove that 1/x+1/y=1/z

Answer 1

Answer:

1/x+1/y=1/z

Step-by-step explanation:

Tip

• parallel lines can be defined as two lines in the same plane that are at equal distance from each other and never meet.

Given

ABllPQllCD

Find

prove that 1/x+1/y=1/z

Solution

In$\triangle \mathrm{ADB} and \triangle \mathrm{PDQ}$.

since $\mathrm{AB} \| \mathrm{PQ}$

$\angle \mathrm{ABQ}=\angle \mathrm{PQD}$ (Corresponding angles)

$\angle \mathrm{ADB}=\angle \mathrm{PDQ} (Common)$

By AA similarity criterion,

$\triangle \mathrm{ADB} \sim \triangle \mathrm{PDQ}\\\therefore \frac{\mathrm{DQ}}{\mathrm{DB}}=\frac{\mathrm{PQ}}{\mathrm{AB}}$

$\Rightarrow \frac{\mathrm{DQ}}{\mathrm{DB}}=\frac{\mathrm{z}}{\mathrm{x}} \quad \ldots \text { (i) }$

Similarly, $\triangle \mathrm{PBQ} \sim \triangle \mathrm{CBD}$

And, $\frac{\mathrm{BQ}}{\mathrm{DB}}=\frac{\mathrm{z}}{\mathrm{x}}$

Adding (i) and (ii), we get

$\frac{z}{x}+\frac{z}{y}=\frac{D Q+B Q}{D B}=\frac{B D}{B D}$

$\Rightarrow \frac{z}{x}+\frac{z}{y}=1$

$\Rightarrow \frac{1}{x}+\frac{1}{y}=\frac{1}{z}$

Final Answer

1/x+1/y=1/z

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