Let $ABC$ be an equilateral triangle with side length $a$. Let $R$ and $r$ denote the radii of the circumcircle and the incircle of triangle $ABC$ respectively. Then,as a function of $a$,the ratio $\frac{R}{r}$
- Astrictly increases
- Bstrictly decreases
- Cremains constant
- Dstrictly increases for $a < 1$ and strictly decreases for $a > 1$
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Similar Questions
In a $\triangle ABC$,let $a, b, c, s, r, R, I, S, r_1, r_2, r_3$ stand for their usual meanings. Match the items of List-$I$ with those of List-$II$.
List-$I$ List-$II$ $A. \tan \frac{A}{2} = \frac{r}{s-a}$ $I. (AI) \left( \frac{\sqrt{(s-b)(s-c)}}{bc} \right)$ $B. r$ $II. R^2$ $C. (SI)^2 + 2Rr$ $III. (4R + r + \sqrt{2}s)(4R + r - \sqrt{2}s)$ $D. r_1^2 + r_2^2 + r_3^2$ $IV. \frac{Rr}{S}$ $V. \frac{(s-b)(s-c)}{\Delta}$
The correct match is:
| List-$I$ | List-$II$ |
| $A. \tan \frac{A}{2} = \frac{r}{s-a}$ | $I. (AI) \left( \frac{\sqrt{(s-b)(s-c)}}{bc} \right)$ |
| $B. r$ | $II. R^2$ |
| $C. (SI)^2 + 2Rr$ | $III. (4R + r + \sqrt{2}s)(4R + r - \sqrt{2}s)$ |
| $D. r_1^2 + r_2^2 + r_3^2$ | $IV. \frac{Rr}{S}$ |
| $V. \frac{(s-b)(s-c)}{\Delta}$ |
The correct match is:
If $r$ is the inradius,$\Delta$ is the area of $\triangle ABC$,and $s$ is the semi-perimeter,then which of the following is true?
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