Question and Answers Forum

GeometryQuestion and Answers: Page 115

Question Number 4077 Answers: 0 Comments: 4

For two co-planer circles to be tangent necessary and sufficient condition is, I think the distance between the centers of circles must be equal to r_1 +r_2 or ∣r_1 −r_2 ∣ , where r_1 and r_2 are the radii of the circles. If circles belong to different planes, what is necessary and sufficient condition for being tangent?

$For two co - planer circles to be tangent$ $necessary and sufficient condition is,$ $I think$ $the distance between the centers of$ $circles must be equal to r_{1} + r_{2} or ∣ r_{1} - r_{2} ∣,$ $where r_{1} and r_{2} are the radii of the circles .$ $If circles belong to different planes, what$ $is necessary and sufficient condition for$ $being tangent ?$

Question Number 4033 Answers: 0 Comments: 15

One circle in a plane can produce one closed region at most(It produces one closed region at least).Two circles in a plane can produce at most three regions(They produce at least two regions).Three circles can produce seven closed regions at most(They produce three closed regions at least). How many distinct closed regions could be produced by 4, 5 and n circles at most? If n circles can produce f(n) closed regions at most(Of course they produce n closed regions at least), can they produce any number of closed regions between n and f(n)?

$One circle in a plane can produce one closed$ $region at most (It produces one closed region$ $at least) . Two circles in a plane can produce$ $at most three regions (They produce at least$ $two regions) . Three circles can produce seven$ $closed regions at most (They produce three$ $closed regions at least) .$ $How many distinct closed regions could be produced$ $by 4, 5 and n circles at most ?$ $If n circles can produce f (n) closed regions$ $at most (Of course they produce n closed$ $regions at least), can they produce any$ $number of closed regions between n and f (n) ?$

Question Number 3950 Answers: 1 Comments: 0

We can make a cyllinder from a rectangle by connecting its opposite edges. Suppose we have two copies of a non−square rectangle.From one copy we make a long cyllinder by connecting long edges of it whereas from other copy by connecting short edges a short cyllinder is made. Compare the volumes of these two cyllinders.

$W e c a n m a k e a c y l l i n d e r f r o m a$ $r e c t a n g l e b y c o n n e c t i n g i t s o p p o s i t e$ $e d g e s .$ $S u p p o s e w e h a v e t w o c o p i e s o f a$ $n o n - s q u a r e r e c t a n g l e . F r o m$ $o n e c o p y w e m a k e a l o n g c y l l i n d e r$ $b y c o n n e c t i n g l o n g e d g e s o f i t$ $w h e r e a s f r o m o t h e r c o p y b y c o n n e c t i n g$ $s h o r t e d g e s a s h o r t c y l l i n d e r i s m a d e .$ $C o m p a r e t h e v o l u m e s o f t h e s e t w o$ $c y l l i n d e r s .$

Question Number 3944 Answers: 0 Comments: 0

Prove that, inside a given square, a semicircle of the largest possible area, can be constructed using ruler and compass.

$P r o v e t h a t, i n s i d e a g i v e n s q u a r e, a s e m i c i r c l e$ $o f t h e l a r g e s t p o s s i b l e a r e a, c a n b e c o n s t r u c t e d$ $u s i n g r u l e r a n d c o m p a s s .$

Question Number 3943 Answers: 0 Comments: 11

What is the area of overlapping region of three circles of radii r_1 , r_2 , r_3 with their respective centres C_1 , C_2 and C_3 when r_1 +r_2 > C_1 C_2 , r_2 +r_3 > C_2 C_3 and r_3 +r_1 > C_3 C_1 . Note that C_i C_j is the distance between centres C_(i ) and C_j .

$W h a t i s t h e a r e a o f o v e r l a p p i n g r e g i o n$ $o f t h r e e c i r c l e s o f r a d i i r_{1}, r_{2}, r_{3} w i t h t h e i r$ $r e s p e c t i v e c e n t r e s C_{1}, C_{2} a n d C_{3} w h e n$ $r_{1} + r_{2} > C_{1} C_{2}, r_{2} + r_{3} > C_{2} C_{3} a n d$ $r_{3} + r_{1} > C_{3} C_{1} .$ $N o t e t h a t C_{i} C_{j} i s t h e d i s t a n c e b e t w e e n c e n t r e s$ $C_{i} a n d C_{j} .$

Question Number 3930 Answers: 0 Comments: 3

y=((log_e ((x/m)−sa))/r^2 ) yr^2 =log_e ((x/m)−sa) e^(yr^2 ) =(x/m)−sa me^(rry) =x−mas Merry christmas everyone! Let our venture for knowledge continue through to the new year!

$y = \frac{\log_{e} (\frac{x}{m} - s a)}{r^{2}}$ ${y r}^{2} = \log_{e} (\frac{x}{m} - s a)$ $e^{{y r}^{2}} = \frac{x}{m} - s a$ ${m e}^{r r y} = x - m a s$ $Merry christmas everyone!$ $Let our venture for knowledge continue$ $through to the new year!$

Question Number 3900 Answers: 1 Comments: 0

I asked this question a while ago, but I forgot how to solve it: S=∫_0 ^( n) ⌊x⌋dx

$I asked this question a while ago,$ $but I forgot how to solve it :$ $S = \int_{0}^{n} ⌊ x ⌋ d x$

Question Number 3877 Answers: 0 Comments: 3

What is the area of overlapping region of two circles having radii r_1 and r_2 when the distance between their centres is c, given that r_1 +r_2 >c.

$W h a t i s t h e a r e a o f o v e r l a p p i n g$ $r e g i o n o f t w o c i r c l e s h a v i n g r a d i i$ $r_{1} a n d r_{2} w h e n t h e d i s t a n c e b e t w e e n$ $t h e i r c e n t r e s i s c, g i v e n t h a t r_{1} + r_{2} > c .$

Question Number 3850 Answers: 0 Comments: 7

How many dimention/s does the point have?

$H o w m a n y d i m e n t i o n / s d o e s t h e p o i n t h a v e ?$

Question Number 3843 Answers: 1 Comments: 4

Let the side of the following mentioned figures is s: The area of square is s^2 , the 3D area(volume) of a cube is s^3 ,the 4D area/volume of 4D hypercube can be said s^4 and so on. Now if radius is r, the area of circle is 𝛑r^2 , the 3D area(volume) of a sphere is(4/3)𝛑r^3 , what will be the 4D area/volume of 4D sphere and 5D area/volume of 5D sphere?

$L e t t h e s i d e o f t h e f o l l o w i n g m e n t i o n e d$ $f i g u r e s i s s :$ $T h e a r e a o f s q u a r e i s s^{2}, t h e 3 D a r e a (v o l u m e)$ $o f a c u b e i s s^{3}, t h e 4 D a r e a / v o l u m e o f 4 D$ $h y p e r c u b e c a n b e s a i d s^{4} a n d s o o n .$ $N o w i f r a d i u s i s r, t h e a r e a o f c i r c l e i s π r^{2},$ $t h e 3 D a r e a (v o l u m e) o f a s p h e r e i s \frac{4}{3} π r^{3},$ $w h a t w i l l b e t h e 4 D a r e a / v o l u m e o f 4 D$ $s p h e r e a n d 5 D a r e a / v o l u m e o f 5 D s p h e r e ?$

Question Number 3840 Answers: 1 Comments: 0

A semicircle contains a square of possible largest area.If s is the measure of the side of the square,what is the radius of the semicircle?

$A s e m i c i r c l e c o n t a i n s a s q u a r e o f$ $p o s s i b l e l a r g e s t a r e a . I f s i s t h e m e a s u r e$ $o f t h e s i d e o f t h e s q u a r e, w h a t i s t h e$ $r a d i u s o f t h e s e m i c i r c l e ?$

Question Number 3838 Answers: 0 Comments: 4

Show that the construction of the rectangle of minimum perimeter when its area is ab where a=AB and b=CD are given is possible with ruler and compass.

$S h o w t h a t t h e c o n s t r u c t i o n o f$ $the rectangle of minimum$ $perimeter when its area is ab$ $where a = AB and b = CD are given$ $i s p o s s i b l e w i t h r u l e r a n d c o m p a s s .$

Question Number 3836 Answers: 0 Comments: 5

A square,whose area is s^2 ,contains a semicircle of possible largest area. Determine radius of the semicircle.

$A s q u a r e, w h o s e a r e a i s s^{2}, c o n t a i n s$ $a s e m i c i r c l e o f p o s s i b l e l a r g e s t a r e a .$ $D e t e r m i n e r a d i u s o f t h e s e m i c i r c l e .$

Question Number 3830 Answers: 0 Comments: 3

Draw a rectangle of maximum perimeter, by ruler and compass,when area is ab. (AB =a,CD=b are given.)

$D r a w a r e c t a n g l e o f m a x i m u m p e r i m e t e r,$ $b y r u l e r a n d c o m p a s s, w h e n a r e a i s ab .$ $(AB = a, CD = b are given .)$

Question Number 3823 Answers: 0 Comments: 2

Consider a triangle ABC. Let D and E are two points on AB and AC respectively such that DE ∥ BC. Now there are two parts of △ABC : △ADE and trapizoid DBCE. If these two regions have same area What will be the ratio of two distances : (i) distance of DE from point A and (ii) distance between BC and DE ?

$C o n s i d e r a t r i a n g l e ABC . L e t D a n d E$ $a r e t w o p o i n t s o n AB a n d AC r e s p e c t i v e l y$ $s u c h t h a t DE ∥ BC . N o w t h e r e a r e t w o$ $p a r t s o f △ ABC : △ ADE a n d t r a p i z o i d$ $DBCE . I f t h e s e t w o r e g i o n s h a v e s a m e a r e a$ $W h a t w i l l b e t h e r a t i o o f t w o d i s t a n c e s :$ $(i) d i s t a n c e o f DE f r o m p o i n t A a n d$ $(i i) d i s t a n c e b e t w e e n BC a n d DE ?$

Question Number 3808 Answers: 2 Comments: 1

A chord divides the circle in two segments,having areas s_1 and s_2 . If diameter, perpendicular to this chord is cut into 1:3 by the chord ,what is s_1 :s_2 ?

$A c h o r d d i v i d e s t h e c i r c l e i n t w o$ $s e g m e n t s, h a v i n g a r e a s s_{1} a n d s_{2} .$ $I f d i a m e t e r, p e r p e n d i c u l a r t o t h i s$ $c h o r d i s c u t i n t o 1 : 3 b y t h e c h o r d, w h a t i s s_{1} : s_{2} ?$

Question Number 3714 Answers: 1 Comments: 0

Derive a formula of volume of right circular cone when the formula of volume of cyllinder is given.

$D erive a formula of volume of right circular cone$ $when the formula of volume of cyllinder is given .$

Question Number 3695 Answers: 0 Comments: 12

Can we say that A line is a circle whose radius is ∞ Or A circle with ∞ radius is a line ?

$C a n w e s a y t h a t$ $A l i n e i s a c i r c l e w h o s e r a d i u s i s \infty$ $O r$ $A c i r c l e w i t h \infty r a d i u s i s a l i n e ?$

Question Number 3662 Answers: 0 Comments: 11

Lets say we have an n−gon. All sides are equal. When n=3, interior angles θ=((180)/3) θ=60° n=4, θ=((360)/4)=90° ⋮ n=t, θ=((180(t−2))/t) For a circle (essentially an ∞−gon): n=∞ ∴θ=180lim_(t→∞) ((t−2)/t) θ=180°????

$Lets say we have an n - gon .$ $All sides are equal .$ $When n = 3, interior angles θ = \frac{180}{3}$ $θ = 60 °$ $n = 4, θ = \frac{360}{4} = 90 °$ $⋮$ $n = t, θ = \frac{180 (t - 2)}{t}$ $For a circle (essentially an \infty - g o n) :$ $n = \infty$ $∴ θ = 180 \lim_{t \to \infty} \frac{t - 2}{t}$ $θ = 180 ° ? ? ? ?$

Question Number 3647 Answers: 0 Comments: 2

Draw a line segment equal to ab units when AB=a units and CD=b units are given. Only ruler and compass may be used.

$D r a w a l i n e s e g m e n t e q u a l t o$ $a b u n i t s w h e n A B = a u n i t s a n d C D = b u n i t s$ $a r e g i v e n . O n l y r u l e r a n d c o m p a s s m a y b e$ $u s e d .$

Question Number 3607 Answers: 1 Comments: 0

Construct a line segment of a^2 units using ruler and compass only, when a line segment of a units is given.

$C o n s t r u c t a l i n e s e g m e n t o f a^{2} u n i t s$ $u s i n g r u l e r a n d c o m p a s s o n l y, w h e n$ $a l i n e s e g m e n t o f a u n i t s i s g i v e n .$

Question Number 3348 Answers: 1 Comments: 3

Prove that the regular pentagon is possible with ruler and compass.

$P r o v e t h a t t h e r e g u l a r p e n t a g o n$ $i s p o s s i b l e w i t h r u l e r a n d c o m p a s s .$

Question Number 3280 Answers: 3 Comments: 0

For a triangle with perpandicular height h and base length b, the area of the triangle is given by: A=(1/2)hb Why is this the case? I understand that two identicle triangles can construct a rectangle, so the area is half of the area of its rectangle with lengths and height b and h Is there any other reasoning?

$For a triangle with perpandicular$ $height h and base length b, the$ $area of the triangle is given by :$ $A = \frac{1}{2} h b$ $Why is this the case ?$ $I understand that two identicle triangles$ $can construct a r e c t a n g l e, so the area$ $is half of the area of its rectangle with$ $lengths and height b and h$ $Is there any other reasoning ?$

Question Number 3262 Answers: 0 Comments: 4

Could ^3 (√2) be drawn on numbered line with the help of ruler and compass only?

$C o u l d^{3} \sqrt{2} b e d r a w n o n n u m b e r e d l i n e w i t h$ $t h e h e l p o f r u l e r a n d c o m p a s s o n l y ?$

Question Number 3249 Answers: 1 Comments: 0

How could (√5) be drawn on numbered line using scale and compass only? (Exactly (√5) not its decimal approximation.)

$H o w c o u l d \sqrt{5} b e d r a w n o n n u m b e r e d l i n e u s i n g$ $s c a l e a n d c o m p a s s o n l y ? (E x a c t l y \sqrt{5} n o t i t s d e c i m a l a p p r o x i m a t i o n .)$

Question Number 3304 Answers: 0 Comments: 9

Bring up the topic/challenge started by Filup at the top. Shall we start new topic at the beginning of calendar month? See older post dt 24.11 by Filup

$Bring up the topic / challenge started by Filup at the top .$ $Shall we start new topic at the beginning$ $of calendar month ?$ $See older post dt 24.11 by Filup$

Pg 110 Pg 111 Pg 112 Pg 113 Pg 114 Pg 115 Pg 116 Pg 117 Pg 118 Pg 119

Contact: info@tinkutara.com