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GeometryQuestion and Answers: Page 115

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°????

$$\mathrm{Lets}\:\mathrm{say}\:\mathrm{we}\:\mathrm{have}\:\mathrm{an}\:{n}−\mathrm{gon}. \\ $$$$\mathrm{All}\:\mathrm{sides}\:\mathrm{are}\:\mathrm{equal}. \\ $$$$ \\ $$$$\mathrm{When}\:{n}=\mathrm{3},\:\mathrm{interior}\:\mathrm{angles}\:\theta=\frac{\mathrm{180}}{\mathrm{3}} \\ $$$$\theta=\mathrm{60}° \\ $$$$ \\ $$$${n}=\mathrm{4},\:\theta=\frac{\mathrm{360}}{\mathrm{4}}=\mathrm{90}° \\ $$$$\vdots \\ $$$${n}={t},\:\theta=\frac{\mathrm{180}\left({t}−\mathrm{2}\right)}{{t}} \\ $$$$ \\ $$$$\mathrm{For}\:\mathrm{a}\:\mathrm{circle}\:\left(\mathrm{essentially}\:\mathrm{an}\:\infty−{gon}\right): \\ $$$${n}=\infty \\ $$$$\therefore\theta=\mathrm{180}\underset{{t}\rightarrow\infty} {\mathrm{lim}}\frac{{t}−\mathrm{2}}{{t}} \\ $$$$\theta=\mathrm{180}°???? \\ $$

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.

$${Draw}\:{a}\:{line}\:{segment}\:{equal}\:{to} \\ $$$$\boldsymbol{{ab}}\:{units}\:{when}\:{AB}=\boldsymbol{{a}}\:{units}\:\:{and}\:{CD}=\boldsymbol{{b}}\:{units}\: \\ $$$${are}\:{given}.\:{Only}\:{ruler}\:{and}\:{compass}\:{may}\:{be} \\ $$$${used}. \\ $$

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.

$${Construct}\:{a}\:{line}\:{segment}\:{of}\:\boldsymbol{{a}}^{\mathrm{2}} \:{units}\: \\ $$$${using}\:{ruler}\:{and}\:\:{compass}\:{only},\:{when} \\ $$$${a}\:{line}\:{segment}\:{of}\:\boldsymbol{{a}}\:{units}\:{is}\:{given}. \\ $$

Question Number 3348 Answers: 1 Comments: 3

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

$${Prove}\:{that}\:{the}\:{regular}\:{pentagon} \\ $$$${is}\:{possible}\:{with}\:{ruler}\:{and}\:{compass}. \\ $$

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?

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

Question Number 3262 Answers: 0 Comments: 4

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

$$\mathcal{C}{ould}\:\:^{\mathrm{3}} \sqrt{\mathrm{2}}\:\:{be}\:{drawn}\:{on}\:{numbered}\:{line}\:{with}\: \\ $$$${the}\:{help}\:\:{of}\:\:{ruler}\:{and}\:{compass}\:{only}? \\ $$

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.)

$$\mathcal{H}{ow}\:{could}\:\sqrt{\mathrm{5}}\:\:{be}\:{drawn}\:{on}\:{numbered}\:{line}\:{using} \\ $$$${scale}\:{and}\:{compass}\:{only}?\:\left({Exactly}\:\sqrt{\mathrm{5}}\:{not}\:{its}\:{decimal}\:{approximation}.\right) \\ $$

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

$$\mathrm{Bring}\:\mathrm{up}\:\mathrm{the}\:\mathrm{topic}/\mathrm{challenge}\:\mathrm{started}\:\mathrm{by}\:\mathrm{Filup}\:\mathrm{at}\:\mathrm{the}\:\mathrm{top}. \\ $$$$\mathrm{Shall}\:\mathrm{we}\:\mathrm{start}\:\mathrm{new}\:\mathrm{topic}\:\mathrm{at}\:\mathrm{the}\:\mathrm{beginning} \\ $$$$\mathrm{of}\:\mathrm{calendar}\:\mathrm{month}? \\ $$$$\mathrm{See}\:\mathrm{older}\:\mathrm{post}\:\mathrm{dt}\:\mathrm{24}.\mathrm{11}\:\mathrm{by}\:\mathrm{Filup} \\ $$

Question Number 3055 Answers: 2 Comments: 0

Given a set S={a_1 , ..., a_n } a_k ∈Z a_1 =a, a_n =a_(n−1) +1, a_n =b ∴S contains all integers between a and b. Are there more even or odd values?

$$\mathrm{Given}\:\mathrm{a}\:\mathrm{set}\:\:{S}=\left\{{a}_{\mathrm{1}} ,\:...,\:{a}_{{n}} \right\} \\ $$$${a}_{{k}} \in\mathbb{Z} \\ $$$$ \\ $$$${a}_{\mathrm{1}} ={a},\:\:\:{a}_{{n}} ={a}_{{n}−\mathrm{1}} +\mathrm{1},\:\:\:{a}_{{n}} ={b} \\ $$$$ \\ $$$$\therefore{S}\:\mathrm{contains}\:\mathrm{all}\:\mathrm{integers}\:\mathrm{between} \\ $$$${a}\:\mathrm{and}\:{b}. \\ $$$$ \\ $$$$\mathrm{Are}\:\mathrm{there}\:\mathrm{more}\:\mathrm{even}\:\mathrm{or}\:\mathrm{odd}\:\mathrm{values}? \\ $$

Question Number 2795 Answers: 2 Comments: 0

Prove that (1+x+x^2 +...)(1+2x+3x^2 +...) =(1/2)(1.2+2.3x+3.4x^2 +...)

$${Prove}\:{that} \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} +...\right)\left(\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +...\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}.\mathrm{2}+\mathrm{2}.\mathrm{3}{x}+\mathrm{3}.\mathrm{4}{x}^{\mathrm{2}} +...\right) \\ $$

Question Number 2499 Answers: 1 Comments: 2

ABCD.EFGH is cube, X is midpoint EF if AB = 6 cm, how distance AX to BD??

$${ABCD}.{EFGH}\:{is}\:{cube},\:{X}\:{is}\:{midpoint}\:{EF} \\ $$$${if}\:{AB}\:=\:\mathrm{6}\:{cm},\:{how}\:{distance}\:{AX}\:{to}\:{BD}?? \\ $$

Question Number 2466 Answers: 3 Comments: 0

(√(−(1/5) ))−(1/(√(−5)))=?

$$\sqrt{−\frac{\mathrm{1}}{\mathrm{5}}\:}−\frac{\mathrm{1}}{\sqrt{−\mathrm{5}}}=? \\ $$

Question Number 2458 Answers: 1 Comments: 0

The medians of a triangle are m_1 , m_2 , m_3 . Find the length of each sides the triangle.

$${The}\:{medians}\:{of}\:{a}\:{triangle} \\ $$$${are}\:{m}_{\mathrm{1}} ,\:{m}_{\mathrm{2}} ,\:{m}_{\mathrm{3}} . \\ $$$${Find}\:{the}\:{length}\:{of}\:{each}\:{sides}\: \\ $$$${the}\:{triangle}. \\ $$

Question Number 2329 Answers: 0 Comments: 3

a_n =(−1)^n (⌊(n/9)⌋+1) Σ_(n≥0) a_n =? b_n =(((a_n −1)(9−a_n ))/(n+1)) Σ_(n≥0) b_n =?

$${a}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \left(\lfloor\frac{{n}}{\mathrm{9}}\rfloor+\mathrm{1}\right) \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}{a}_{{n}} =? \\ $$$${b}_{{n}} =\frac{\left({a}_{{n}} −\mathrm{1}\right)\left(\mathrm{9}−{a}_{{n}} \right)}{{n}+\mathrm{1}} \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}{b}_{{n}} =? \\ $$

Question Number 2196 Answers: 1 Comments: 0

(a+b)^2 =a^2 +b^2 +2ab (a+b+c)^2 =a^2 +b^2 +c^2 +2ab+2bc+2ca (a+b+c+d)^2 =? (a_1 +a_2 +...+a_n )^2 =? Derive a formula or give a technique.

$$\left({a}+{b}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{2}{ab} \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\mathrm{2}{ab}+\mathrm{2}{bc}+\mathrm{2}{ca} \\ $$$$\left({a}+{b}+{c}+{d}\right)^{\mathrm{2}} =? \\ $$$$\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +...+{a}_{{n}} \right)^{\mathrm{2}} =? \\ $$$${Derive}\:{a}\:{formula}\:{or}\:{give}\:{a}\:{technique}. \\ $$

Question Number 2138 Answers: 0 Comments: 8

Is the following proof correct? Δ=Σ_(i=0) ^∞ (−1)^i 2^i =1−2+4−8+16−32+... Let: Δ_1 =1−2+4−8+16−32+... Δ_2 = 1−2+4−8+16−32+... Δ_1 +Δ_2 =1+(−2+1)+(4−2)+(−8+4)+... ∴Δ_1 +Δ_2 =1−(1−2+4−8+...) ∴Δ_1 +Δ_2 =1−Δ_1 Δ_1 =Δ_2 =Δ 3Δ=1 ∴Δ=(1/3) ∴Δ=Σ_(i=0) ^∞ (−1)^i 2^i =(1/3)

$$\mathrm{Is}\:\mathrm{the}\:\mathrm{following}\:\mathrm{proof}\:\mathrm{correct}? \\ $$$$ \\ $$$$\Delta=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{i}} \mathrm{2}^{{i}} =\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+... \\ $$$$ \\ $$$$\mathrm{Let}: \\ $$$$\Delta_{\mathrm{1}} =\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+... \\ $$$$\Delta_{\mathrm{2}} =\:\:\:\:\:\:\:\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+... \\ $$$$ \\ $$$$\Delta_{\mathrm{1}} +\Delta_{\mathrm{2}} =\mathrm{1}+\left(−\mathrm{2}+\mathrm{1}\right)+\left(\mathrm{4}−\mathrm{2}\right)+\left(−\mathrm{8}+\mathrm{4}\right)+... \\ $$$$\therefore\Delta_{\mathrm{1}} +\Delta_{\mathrm{2}} =\mathrm{1}−\left(\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+...\right) \\ $$$$\therefore\Delta_{\mathrm{1}} +\Delta_{\mathrm{2}} =\mathrm{1}−\Delta_{\mathrm{1}} \\ $$$$\Delta_{\mathrm{1}} =\Delta_{\mathrm{2}} =\Delta \\ $$$$\mathrm{3}\Delta=\mathrm{1} \\ $$$$\therefore\Delta=\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$ \\ $$$$\therefore\Delta=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{i}} \mathrm{2}^{{i}} =\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$ \\ $$

Question Number 1999 Answers: 1 Comments: 0

if (1 + tan 1°)(1 + tan 2°)(1 + tan 3°)...... ....)(1 + tan 44°)(1 + tan 45°) ^ = { ((((√(50)) + 7)))^(1/3) − ((((√(50))−7)))^(1/3) }^((x − 7)) find x = ...?

$${if}\: \\ $$$$\left(\mathrm{1}\:+\:{tan}\:\mathrm{1}°\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{2}°\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{3}°\right)...... \\ $$$$\left.....\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{44}°\right)\left(\mathrm{1}\:+\:{tan}\:\mathrm{45}°\right)\overset{} {\:}=\:\:\left\{\:\:\:\sqrt[{\mathrm{3}}]{\left(\sqrt{\mathrm{50}}\:+\:\mathrm{7}\right)}\:−\:\sqrt[{\mathrm{3}}]{\left(\sqrt{\mathrm{50}}−\mathrm{7}\right)}\:\overset{\left({x}\:−\:\mathrm{7}\right)} {\right\}} \\ $$$$\: \\ $$$${find}\:{x}\:=\:...? \\ $$

Question Number 2001 Answers: 0 Comments: 0

Prove that : (1/(15)) < (1/2) ∙ (3/4) ∙ (5/6) ∙ ∙ ∙ ∙ ∙ ((99)/(100)) < (1/(10))

$${Prove}\:{that}\:: \\ $$$$\frac{\mathrm{1}}{\mathrm{15}}\:<\:\frac{\mathrm{1}}{\mathrm{2}}\:\centerdot\:\frac{\mathrm{3}}{\mathrm{4}}\:\centerdot\:\frac{\mathrm{5}}{\mathrm{6}}\:\centerdot\:\centerdot\:\centerdot\:\centerdot\:\centerdot\:\frac{\mathrm{99}}{\mathrm{100}}\:<\:\frac{\mathrm{1}}{\mathrm{10}} \\ $$

Question Number 1928 Answers: 2 Comments: 10

Prove that, if p>q>0 and x≥0, then (1/p)((x^p /(p+1))−1)≥(1/q)((x^q /(q+1))−1).

$${Prove}\:{that},\:{if}\:{p}>{q}>\mathrm{0}\:{and}\:{x}\geqslant\mathrm{0},\:{then} \\ $$$$\:\:\:\:\:\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right).\: \\ $$

Question Number 1793 Answers: 0 Comments: 0

Evaluate ∫_(1/2) ^2 ((sinx)/(x(sinx+sin(1/x))))dx .

$${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{1}/\mathrm{2}} ^{\mathrm{2}} \frac{{sinx}}{{x}\left({sinx}+{sin}\frac{\mathrm{1}}{{x}}\right)}{dx}\:. \\ $$

Question Number 1740 Answers: 1 Comments: 1

how 0!=1

$${how}\:\:\mathrm{0}!=\mathrm{1} \\ $$

Question Number 1750 Answers: 1 Comments: 0

(x+3)^3

$$\left({x}+\mathrm{3}\right)^{\mathrm{3}} \\ $$

Question Number 1611 Answers: 0 Comments: 1

A right angled triangle has fixed hypotenuse measuring h units. What are the measures of its legs, for maximum perimeter P units. Will the area be also maximum, when the perimeter be maximum?

$$\mathrm{A}\:\mathrm{right}\:\mathrm{angled}\:\mathrm{triangle}\:\mathrm{has}\:\mathrm{fixed}\:\mathrm{hypotenuse}\:\mathrm{measuring} \\ $$$$\mathrm{h}\:\mathrm{units}.\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{measures}\:\:\mathrm{of}\:\mathrm{its}\:\mathrm{legs}, \\ $$$$\mathrm{for}\:\boldsymbol{\mathrm{maximum}}\:\boldsymbol{\mathrm{perimeter}}\:\mathrm{P}\:\mathrm{units}. \\ $$$$\mathrm{Will}\:\mathrm{the}\:\mathrm{area}\:\mathrm{be}\:\mathrm{also}\:\mathrm{maximum},\:\mathrm{when}\:\mathrm{the}\:\mathrm{perimeter}\:\mathrm{be} \\ $$$$\mathrm{maximum}? \\ $$

Question Number 1592 Answers: 1 Comments: 1

I have a loop of string of length(perimeter) p units. I want to make a triangle of largest area from the loop. What will be the dimensions of that triangle?

$$\mathrm{I}\:\mathrm{have}\:\mathrm{a}\:\mathrm{loop}\:\mathrm{of}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length}\left(\mathrm{perimeter}\right)\:\:\mathrm{p}\:\mathrm{units}.\: \\ $$$$\mathrm{I}\:\mathrm{want}\:\mathrm{to}\:\mathrm{make}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{of}\:\mathrm{largest}\:\mathrm{area}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{loop}.\:\mathrm{What}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{dimensions}\:\mathrm{of}\:\mathrm{that}\:\mathrm{triangle}? \\ $$

Question Number 1581 Answers: 0 Comments: 2

Find a function f(x) satisfying the following equation. ∫_a ^( b) [(1/2){f(x)}^2 −(√({f(x)}^2 +{(d/dx)(f(x))}^2 ))]dx=0 b>0,a>0 , b≠a.

$${Find}\:{a}\:{function}\:{f}\left({x}\right)\:{satisfying} \\ $$$${the}\:{following}\:{equation}. \\ $$$$\int_{{a}} ^{\:{b}} \left[\frac{\mathrm{1}}{\mathrm{2}}\left\{{f}\left({x}\right)\right\}^{\mathrm{2}} −\sqrt{\left\{{f}\left({x}\right)\right\}^{\mathrm{2}} +\left\{\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)\right\}^{\mathrm{2}} }\right]{dx}=\mathrm{0} \\ $$$${b}>\mathrm{0},{a}>\mathrm{0}\:,\:{b}\neq{a}.\:\: \\ $$

Question Number 1387 Answers: 0 Comments: 1

Q. is there any angle in a circle?

$${Q}.\:{is}\:{there}\:{any}\:{angle}\:{in}\:{a}\:{circle}? \\ $$

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