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Question Number 16874 Answers: 0 Comments: 0

Let ABC be an acute triangle. The interior bisectors of the angles ∠B and ∠C meet the opposite sides at the points L and M, respectively. Prove that there exists a point K in the interior of the side BC such that ΔKLM is equilateral if and only if ∠A = 60°.

$$\mathrm{Let}\:{ABC}\:\mathrm{be}\:\mathrm{an}\:\mathrm{acute}\:\mathrm{triangle}.\:\mathrm{The} \\ $$$$\mathrm{interior}\:\mathrm{bisectors}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angles}\:\angle{B}\:\mathrm{and} \\ $$$$\angle{C}\:\mathrm{meet}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{sides}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{points}\:{L}\:\mathrm{and}\:{M},\:\mathrm{respectively}.\:\mathrm{Prove} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{point}\:{K}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{interior}\:\mathrm{of}\:\mathrm{the}\:\mathrm{side}\:{BC}\:\mathrm{such}\:\mathrm{that} \\ $$$$\Delta{KLM}\:\mathrm{is}\:\mathrm{equilateral}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if} \\ $$$$\angle{A}\:=\:\mathrm{60}°. \\ $$

Question Number 16911 Answers: 0 Comments: 1

Please solve Q. 16066 or please tell me whether to post its solution or not. I don′t understood the solution.

$$\mathrm{Please}\:\mathrm{solve}\:\mathrm{Q}.\:\mathrm{16066}\:\mathrm{or}\:\mathrm{please}\:\mathrm{tell}\:\mathrm{me} \\ $$$$\mathrm{whether}\:\mathrm{to}\:\mathrm{post}\:\mathrm{its}\:\mathrm{solution}\:\mathrm{or}\:\mathrm{not}.\:\mathrm{I} \\ $$$$\mathrm{don}'\mathrm{t}\:\mathrm{understood}\:\mathrm{the}\:\mathrm{solution}. \\ $$

Question Number 16873 Answers: 0 Comments: 0

Let I be the incenter of ΔABC. It is known that for every point M ∈ (AB), one can find the points N ∈ (BC) and P ∈ (AC) such that I is the centroid of ΔMNP. Prove that ABC is an equilateral triangle.

$$\mathrm{Let}\:{I}\:\mathrm{be}\:\mathrm{the}\:\mathrm{incenter}\:\mathrm{of}\:\Delta{ABC}.\:\mathrm{It}\:\mathrm{is} \\ $$$$\mathrm{known}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\:\mathrm{point}\:{M}\:\in\:\left({AB}\right), \\ $$$$\mathrm{one}\:\mathrm{can}\:\mathrm{find}\:\mathrm{the}\:\mathrm{points}\:{N}\:\in\:\left({BC}\right)\:\mathrm{and} \\ $$$${P}\:\in\:\left({AC}\right)\:\mathrm{such}\:\mathrm{that}\:{I}\:\mathrm{is}\:\mathrm{the}\:\mathrm{centroid}\:\mathrm{of} \\ $$$$\Delta{MNP}.\:\mathrm{Prove}\:\mathrm{that}\:{ABC}\:\mathrm{is}\:\mathrm{an} \\ $$$$\mathrm{equilateral}\:\mathrm{triangle}. \\ $$

Question Number 16868 Answers: 1 Comments: 0

In how many ways can the letters of the word. EVERMORE be arrange if the word must begin with (i) R (ii) E

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\:\mathrm{word}.\:\mathrm{EVERMORE}\:\mathrm{be}\:\mathrm{arrange} \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{word}\:\mathrm{must}\:\mathrm{begin}\:\mathrm{with}\: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{R} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{E} \\ $$

Question Number 16859 Answers: 0 Comments: 0

In dealing with motion of projectile in air, we ignore effect of air resistance on motion. What would the trajectory look like if air resistance is included? Sketch such a trajectory and explain why you have drawn it that way.

$$\mathrm{In}\:\mathrm{dealing}\:\mathrm{with}\:\mathrm{motion}\:\mathrm{of}\:\mathrm{projectile}\:\mathrm{in} \\ $$$$\mathrm{air},\:\mathrm{we}\:\mathrm{ignore}\:\mathrm{effect}\:\mathrm{of}\:\mathrm{air}\:\mathrm{resistance} \\ $$$$\mathrm{on}\:\mathrm{motion}.\:\mathrm{What}\:\mathrm{would}\:\mathrm{the}\:\mathrm{trajectory} \\ $$$$\mathrm{look}\:\mathrm{like}\:\mathrm{if}\:\mathrm{air}\:\mathrm{resistance}\:\mathrm{is}\:\mathrm{included}? \\ $$$$\mathrm{Sketch}\:\mathrm{such}\:\mathrm{a}\:\mathrm{trajectory}\:\mathrm{and}\:\mathrm{explain} \\ $$$$\mathrm{why}\:\mathrm{you}\:\mathrm{have}\:\mathrm{drawn}\:\mathrm{it}\:\mathrm{that}\:\mathrm{way}. \\ $$

Question Number 16857 Answers: 1 Comments: 0

Suppose x and y are vectors in R^n that have the same length. show that x + y bisect the angle between x and y.

$$\mathrm{Suppose}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}\:\mathrm{are}\:\mathrm{vectors}\:\mathrm{in}\:\mathbb{R}^{\mathrm{n}} \:\mathrm{that}\:\mathrm{have}\:\mathrm{the}\:\mathrm{same}\:\mathrm{length}.\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:\:\mathrm{bisect}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{between}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y}. \\ $$

Question Number 16855 Answers: 1 Comments: 0

If sin θ = (1/2), cos φ = (1/3), then θ + φ belongs to, where 0 < θ, φ < (π/2) (1) ((π/3), (π/2)) (2) ((π/2), ((2π)/3))

$$\mathrm{If}\:\mathrm{sin}\:\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}},\:\mathrm{cos}\:\phi\:=\:\frac{\mathrm{1}}{\mathrm{3}},\:\mathrm{then}\:\theta\:+\:\phi \\ $$$$\mathrm{belongs}\:\mathrm{to},\:\mathrm{where}\:\mathrm{0}\:<\:\theta,\:\phi\:<\:\frac{\pi}{\mathrm{2}} \\ $$$$\left(\mathrm{1}\right)\:\left(\frac{\pi}{\mathrm{3}},\:\frac{\pi}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{2}\right)\:\left(\frac{\pi}{\mathrm{2}},\:\frac{\mathrm{2}\pi}{\mathrm{3}}\right) \\ $$

Question Number 16845 Answers: 1 Comments: 3