prove-that-z-1-z-2-z-1-z-2-

Question Number 10597 by Saham last updated on 19/Feb/17

$$\mathrm{prove}\:\mathrm{that} \\ $$$$\mid\mathrm{z}_{\mathrm{1}} −\:\mathrm{z}_{\mathrm{2}} \mid\:\leqslant\:\mid\mathrm{z}_{\mathrm{1}} \mid\:+\:\mid\mathrm{z}_{\mathrm{2}} \mid \\ $$

Answered by mrW1 last updated on 19/Feb/17

see picture. OB=∣Z_1 +Z_2 ∣ OC=∣Z_1 −Z_2 ∣ OA=∣Z_1 ∣ AB=∣Z_2 ∣ AC=∣−Z_2 ∣=∣Z_2 ∣ ∴ OB≤OA+AB ∵ ∣Z_1 +Z_2 ∣≤∣Z_1 ∣+∣Z_2 ∣ ∴ OC≤OA+AC ∵ ∣Z_1 −Z_2 ∣≤∣Z_1 ∣+∣Z_2 ∣

$${see}\:{picture}. \\ $$$${OB}=\mid{Z}_{\mathrm{1}} +{Z}_{\mathrm{2}} \mid \\ $$$${OC}=\mid{Z}_{\mathrm{1}} −{Z}_{\mathrm{2}} \mid \\ $$$${OA}=\mid{Z}_{\mathrm{1}} \mid \\ $$$${AB}=\mid{Z}_{\mathrm{2}} \mid \\ $$$${AC}=\mid−{Z}_{\mathrm{2}} \mid=\mid{Z}_{\mathrm{2}} \mid \\ $$$$ \\ $$$$\therefore\:{OB}\leqslant{OA}+{AB} \\ $$$$\because\:\mid{Z}_{\mathrm{1}} +{Z}_{\mathrm{2}} \mid\leqslant\mid{Z}_{\mathrm{1}} \mid+\mid{Z}_{\mathrm{2}} \mid \\ $$$$ \\ $$$$\therefore\:{OC}\leqslant{OA}+{AC} \\ $$$$\because\:\mid{Z}_{\mathrm{1}} −{Z}_{\mathrm{2}} \mid\leqslant\mid{Z}_{\mathrm{1}} \mid+\mid{Z}_{\mathrm{2}} \mid \\ $$

Commented by Saham last updated on 19/Feb/17

$$\mathrm{i}\:\mathrm{really}\:\mathrm{appreciate}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

Commented by Saham last updated on 19/Feb/17

can i connect you on whatsapp sir. i can drop my number sir.

$$\mathrm{can}\:\mathrm{i}\:\mathrm{connect}\:\mathrm{you}\:\mathrm{on}\:\mathrm{whatsapp}\:\mathrm{sir}.\:\mathrm{i}\:\mathrm{can}\:\mathrm{drop}\:\mathrm{my} \\ $$$$\mathrm{number}\:\mathrm{sir}. \\ $$

Commented by mrW1 last updated on 20/Feb/17