If-a-flagstaff-subtends-equal-angles-at-4-points-A-B-C-and-D-on-the-horizontal-plane-through-the-foot-of-the-flagstaff-then-A-B-C-and-D-must-be-the-vertices-of-1-Square-2-Cyclic-quadrilateral

FacebookTweetPin

Question Number 15384 by Tinkutara last updated on 10/Jun/17

$$\mathrm{If}\mathrm{a}\mathrm{flagstaff}\mathrm{subtends}\mathrm{equal}\mathrm{angles}\mathrm{at}4$$$$\mathrm{points}A,B,C\mathrm{and}D\mathrm{on}\mathrm{the}\mathrm{horizontal}$$$$\mathrm{plane}\mathrm{through}\mathrm{the}\mathrm{foot}\mathrm{of}\mathrm{the}\mathrm{flagstaff},$$$$\mathrm{then}A,B,C\mathrm{and}D\mathrm{must}\mathrm{be}\mathrm{the}$$$$\mathrm{vertices}\mathrm{of}$$$$\left(1\right)\mathrm{Square}$$$$\left(2\right)\mathrm{Cyclic}\mathrm{quadrilateral}$$$$\left(3\right)\mathrm{Rectangle}$$$$\left(4\right)\mathrm{Parallelogram}$$

Answered by mrW1 last updated on 10/Jun/17

$$\left(2\right)$$

Commented by Tinkutara last updated on 10/Jun/17

$$\mathrm{Sir}\mathrm{please}\mathrm{explain}.$$

Commented by mrW1 last updated on 10/Jun/17

$$\mathrm{h}=\mathrm{height}\mathrm{of}\mathrm{flagstaff}$$$${\mathrm{d}}_{\mathrm{A}}=\mathrm{distance}\mathrm{of}\mathrm{point}\mathrm{A}\mathrm{to}\mathrm{the}\mathrm{foot}\mathrm{of}\mathrm{flagstaff}$$$${\theta }_{\mathrm{A}}=\mathrm{angle}\mathrm{flagstaff}\mathrm{subtends}\mathrm{at}\mathrm{point}\mathrm{A}$$$$\tan {\theta }_{\mathrm{A}}=\frac{\mathrm{h}}{{\mathrm{d}}_{\mathrm{A}}}$$$$\tan {\theta }_{\mathrm{B}}=\frac{\mathrm{h}}{{\mathrm{d}}_{\mathrm{B}}}$$$$\tan {\theta }_{\mathrm{C}}=\frac{\mathrm{h}}{{\mathrm{d}}_{\mathrm{C}}}$$$$\tan {\theta }_{\mathrm{D}}=\frac{\mathrm{h}}{{\mathrm{d}}_{\mathrm{D}}}$$$$\because {\theta }_{\mathrm{A}}={\theta }_{\mathrm{B}}={\theta }_{\mathrm{C}}={\theta }_{\mathrm{D}}$$$$\therefore {\mathrm{d}}_{\mathrm{A}}={\mathrm{d}}_{\mathrm{B}}={\mathrm{d}}_{\mathrm{C}}={\mathrm{d}}_{\mathrm{D}}=\mathrm{r}$$$$\Rightarrow \mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{distance}$$$$\mathrm{to}\mathrm{the}\mathrm{foot}\mathrm{of}\mathrm{flagstaff},\mathrm{or}\mathrm{they}\mathrm{must}$$$$\mathrm{lie}\mathrm{on}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{center}\mathrm{point}\mathrm{at}$$$$\mathrm{the}\mathrm{foot}\mathrm{of}\mathrm{flagstaff}.$$$$\Rightarrow \mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\mathrm{are}\mathrm{vertices}\mathrm{of}\mathrm{a}\mathrm{cyclic}$$$$\mathrm{quadrilateral}.$$

Commented by mrW1 last updated on 10/Jun/17

Commented by Tinkutara last updated on 10/Jun/17

$$\mathrm{Thanks}\mathrm{Sir}!$$

Terms of Service
Privacy Policy
Contact: info@tinkutara.com

FacebookTweetPin