Question

In the figure seg AB and seg DC are perpendicular to seg BC . Seg AC and seg BD intersect each other at P. seg PQ is perpendicular to seg BC.AB= x, PQ=y , DC= z . Prove that 1/x +1/z= 1/y​

Answer 1

$\frac{1}{x} + \frac{1}{z} = \frac{1}{y}$ (Proved)

Step-by-step explanation:

See the diagram attached.

Given AB and CD are perpendicular to BC and PQ is also perpendicular to BC.

So, AB ║ CD ║ PQ.

Now, considering the triangles Δ ABC and Δ PQC, they are similar triangles as AB ║ PQ.

So, $\frac{PQ}{AB} = \frac{QC}{BC}$

⇒ $\frac{y}{x} = \frac{b}{a + b}$ ........... (1)

Again, considering the triangles Δ BCD and Δ BPQ, they are similar triangles as CD ║ PQ.

So, $\frac{PQ}{CD} = \frac{BQ}{BC}$

⇒ $\frac{y}{z} = \frac{a}{a + b}$ ............ (2)

Adding equations (1) and (2) we get,

$\frac{y}{x} + \frac{y}{z} = \frac{a}{a + b} + \frac{b}{a + b} = 1$

⇒ $\frac{1}{x} + \frac{1}{z} = \frac{1}{y}$ (Proved)

Answer 2

hence proced 1/x+1/z=1/y