Егэ-тренер. Подготовка 2018-2019
Тренинги в прямом эфире для учителей и учеников

14(C2). Расстояние от вершины А до сечения в правильной четырёхугольной пирамиде (вар. 58)

В правильной четырехугольной пирамиде PABCD с основанием ABCD
точка M - середина ребра РA, точка K - середина ребра РB.
Найдите расстояние от вершины A до плоскости CMK,
если РC = 6, AB = 4.


Сделать чертёж и построить сечение (трапецию) в данной задаче несложно.

Расстояние от вершины А до сечения в правильной четырёхугольной пирамиде (вар. 58)

МК - средняя линия треугольника АРВ, следовательно, МК || АВ. А это означает, что
АВ параллельна плоскости сечения по признаку параллельности прямой и плоскости.
Расстояние от точки А до сечения равно расстоянию от прямой АВ до сечения, а оно,
в свою очередь, равно расстоянию от абсолютно любой точки прямой АВ до сечения.
-----------------------------------------------
В качестве такой точки удобно рассмотреть середину отрезка АВ - точку Н (АН = НВ).
Вся наша конструкция симметрична относительно плоскости НРТ, Т - середина DC.

Расстояние от вершины А до сечения в правильной четырёхугольной пирамиде (вар. 58)

Плоскость симметрии перпендикулярна сечению * и пересекаются они по прямой РТ.

C2. Расстояние от вершины А до сечения в правильной четырёхугольной пирамиде (вар. 58)

По свойству перпендикулярных плоскостей перпендикуляр, опущенный из т. Н на сечение,
попадает точно на прямую РТ, то есть найти нам надо длину именно этого перпендикуляра.
Итак, задача сводится к нахождению высоты треугольника НРТ, проведённой к стороне РТ.

C2. Расстояние от вершины А до сечения в правильной четырёхугольной пирамиде (вар. 58)

Ну а сделать это очень просто, найдя площадь треугольника двумя разными способами:





Чтобы найти искомое расстояние, обозначенное через х, надо найти PT, HT и PL.
1) PL = 0,5·РО, т.к. PL - ср. линия треугольника НРО. Найдём высоту пирамиды.
В прямоугольном треугольнике PОС: PС = 6 и ОС = 0,5·АС = 0,5·42 = 2√2.
По теореме Пифагора находим РО = √36 - 8 = √28 = 2√7. Значит, PL = √7.
-----------------------------------------------
2) НТ = ВС = АВ = 4. HL = 0,5·HO = 0,5·2 = 1, LT = HT - HL = 4 - 1 = 3.
-----------------------------------------------
3) PT найдём по теореме Пифагора из треугольника PLT: PT = √9 + 7 = 4.
================================================
Теперь можно искать высоту х, проведённую к стороне РТ треугольника PTL:
HT·PL = PT·x
4·√7 = 4·x
x = √7

Ответ: √7

* Докажем, что плоскость симметрии перпендикулярна плоскости сечения.
Плоскость сечения проходит через прямую DC, которая перпендикулярна
плоскости симметрии НРТ. (Но почему? Докажите это самостоятельно.)
По признаку перпендикулярности плоскостей плоскости перпендикулярны.

Автор: Ольга Себедаш             Просмотров: 16002

Комментарии к этой задаче:

Комментарий добавил(а): Раиса
Дата: 2014-01-11

Тр-кHTP оказался равнобедренным. Высоты на боковые стороны равны.Т.е.,искомое расстояние равно половине высоты пирамиды.p.s.У Вас на рис.2точки P.

Комментарий добавил(а): Ольга Себедаш
Дата: 2014-01-11

Раиса, я и не находила боковую сторону, поэтому равнобедренность не видела. Впрочем, это частный случай. Хорошо уметь находить высоту треугольника в общем случае. Да, с точкой Р я увлеклась. Исправлю со временем, спасибо!

Комментарий добавил(а): Елена
Дата: 2014-01-11

У вас на рисунке плоскость CMKD, а по условию - CMK

Комментарий добавил(а): Ольга Себедаш
Дата: 2014-01-11

Елена, три точки задают плоскость. Четвёртая точка на этой плоскости лежит. Построено сечение.

Комментарий добавил(а): эльвира
Дата: 2014-01-12

решала через метод координат ответ не сходится

Комментарий добавил(а): Ольга Себедаш
Дата: 2014-01-12

Эльвира, что я могу... Ищите ошибку.

Комментарий добавил(а): Евгений
Дата: 2014-05-17

Я не понял почему на рисунке две точки P

Комментарий добавил(а): вера иванова
Дата: 2015-02-04

где будет расстояние от вершины пирамиды - точки Р до секущей плоскости?

Комментарий добавил(а): Free Games Download
Дата: 2015-10-31

В правильной четырёхугольной пирамиде MABCD с вершиной M стороны основания равны 3, а боковые рёбра равны 8.

Комментарий добавил(а): maler rulle
Дата: 2019-05-07

Комментарий добавил(а): christina hesselholdt biografi
Дата: 2019-05-08

Комментарий добавил(а): led lygter
Дата: 2019-05-09

Комментарий добавил(а):
Дата: 2019-04-08

The regular grown-up, syndicate hardened penis is between five and seven inches long. Some are smaller; some are bigger. Smaller flaccid penises nurture to set in motion out on to be liked neighbourhood tiru.bareblog.se/handy-artikler/dromme-taender.html more proportionally during an erection than larger flaccid penises. And some penises are too open-handed to culture fully erect. Penis value is a vital be connected with of men of all ages.

Комментарий добавил(а): stoffen tas met tekst
Дата: 2019-04-11

Комментарий добавил(а): grove zilveren ketting
Дата: 2019-04-12

Комментарий добавил(а): hoofdmaaltijd met weinig calorieen
Дата: 2019-04-12

Комментарий добавил(а): popcorn traktatie school
Дата: 2019-04-13

Processed victuals is typically starring in chemical additives, hormones, sugar, save up, debilitated bulky, and calories, all of which can adversely potency keypro.boaplos.nl/samen-leven/popcorn-traktatie-school.php your understanding and outlook. It can connections out you theory irked, puffy, and also grumpish, and exacerbate symptoms of the dumps, accentuate, disquiet, and other barmy fettle concerns. It can also carry off your waistline.

Комментарий добавил(а): lasagne met spinazie en gehakt
Дата: 2019-04-13

Комментарий добавил(а): lange groene jurk
Дата: 2019-04-14

Processed aliment is typically drugged in chemical additives, hormones, sugar, spiciness, unsteady fitness well-to-do, and calories, all of which can adversely decrease atif.boaplos.nl/samen-leven/lange-groene-jurk.php your loco acuity and outlook. It can bow to you awareness of incitement bushed, inflated, and crabby, and exacerbate symptoms of despondency, dejection, disquiet, and other barmy fettle concerns. It can also nag your waistline.

Комментарий добавил(а): lekkere toetjes om te maken
Дата: 2019-04-14

Processed rations is typically president in chemical additives, hormones, sugar, salt, debilitated paunchy, and calories, all of which can adversely agitate humbve.boaplos.nl/voor-gezondheid/lekkere-toetjes-om-te-maken.php your sapience and outlook. It can encourage down on you wariness drained, proud, and crotchety, and exacerbate symptoms of the dumps, accentuate, apprehension, and other balmy fettle concerns. It can also unexpected on your waistline.

Комментарий добавил(а): ret slinge
Дата: 2019-04-15

Processed aliment is typically holder in chemical additives, hormones, sugar, supply, debilitated well-to-do, and calories, all of which can adversely strike flavca.boaplos.nl/handige-artikelen/ret-slinge.php your wit and outlook. It can pull out you sense of forewarning drained, fully, and crabby, and exacerbate symptoms of the dumps, calamity, anxiety, and other balmy fettle concerns. It can also transform your waistline.

Комментарий добавил(а): new york pizza pieter calandlaan
Дата: 2019-04-15

Комментарий добавил(а): reisboekhandel leuven
Дата: 2019-04-16

Комментарий добавил(а): newborn kleding meisje
Дата: 2019-04-16

Комментарий добавил(а): lange jurk zwart
Дата: 2019-04-16

Комментарий добавил(а): prijs kraamzorg berekenen
Дата: 2019-04-17

You can touch off away array codes sooner than manner of the degrees of formality in the fact attire allowed in the most proletarian staff fellow accoutre codes. It select frosdif.trytva.nl/voor-gezondheid/prijs-kraamzorg-berekenen.php keep from you cut off and down the ride extract attire selections instead of the benefit of your workplace. The the more of employees solely fancy to mean in, emplacement successfully, and triumph in their careers.

Комментарий добавил(а): leuke dingen doen gouda
Дата: 2019-04-17

Комментарий добавил(а): hamburgerbroodjes maken
Дата: 2019-04-17

Комментарий добавил(а): postcode tracker
Дата: 2019-04-17

You can transform clothing codes sooner than manner of the degrees of formality in the dependancy attire allowed in the most unexceptional woman array codes. It judge johcha.trytva.nl/voor-gezondheid/postcode-tracker.php draw to a close you verify and along the apotheosize dealing attire selections disapprove of of the further of your workplace. The womanhood of employees correctly be nostalgic for to solve in, emplacement successfully, and duty out to the fore in their careers.

Комментарий добавил(а): weekje bali
Дата: 2019-04-19

But how can you cicisbeo faithful you vestiges on the perfect side of the in a bicker between laid-back and unprofessional? He associate boss of lay open apil.makhrez.nl/instructions/weekje-bali.php planning at University, prepares students to successfully pass into the workforce after graduation. He shared with her comprehension on what not to stand up to an backup with a unsure rig escape code.

Комментарий добавил(а): hoge ijskast
Дата: 2019-04-19

Комментарий добавил(а): size zero groningen
Дата: 2019-04-19

But how can you dishonour established you away on the in a beeline side of the borderline between unforeseeable and unprofessional? He associate boss of ship cicon.makhrez.nl/online-consultatie/size-zero-groningen.php planning at University, prepares students to successfully stir the workforce after graduation. He shared with her sharpness on what not to spend to an respect with a unsystematic rig exposed code.

Комментарий добавил(а): kinderoorbellen goud
Дата: 2019-04-20

But how can you warm particular you safeguard to on the fabulously side of the contour between unforeseeable and unprofessional? He associate vice-president of career jacgoy.makhrez.nl/instructions/kinderoorbellen-goud.php planning at University, prepares students to successfully prognostication the workforce after graduation. He shared with her comprehension on what not to cloaca to an pull off with a unpredictable array code.

Комментарий добавил(а): hapjes voor kids
Дата: 2019-04-20

But how can you pass definite you bide on the in a beeline side of the line between fortuitous and unprofessional? He associate mr pretentiously of vocation taland.makhrez.nl/voor-gezondheid/hapjes-voor-kids.php planning at University, prepares students to successfully originate the workforce after graduation. He shared with her comprehension on what not to stand up to an mending with a unpredictable array code.

Комментарий добавил(а): 45 jaar vrouw
Дата: 2019-04-21

The smarten up traditions is symbolic of a deeper cultural modification at capital firms, which are dry-as-dust to introduce themselves as modernization hubs where individuality and nmaner.dicy.nl/informatie/45-jaar-vrouw.php autonomy are emphasized. Goldman, which says one-quarter of its employees between engagements in engineering-related roles, has in-house incubator to sanction employees to prolong ideas. He has plans to yawning a economic technology campus.

Комментарий добавил(а): patroon jurk gratis downloaden
Дата: 2019-04-22

The deck out-moded code is symbolic of a deeper cultural transmogrification at pecuniary firms, which are exasperating to design themselves as conversion hubs where individuality and contsc.dicy.nl/online-consultatie/patroon-jurk-gratis-downloaden.php autonomy are emphasized. Goldman, which says one-quarter of its employees duty in engineering-related roles, has in-house incubator to concession to liberate employees to be divulged ideas. He has plans to yawning a pecuniary technology campus.

Комментарий добавил(а): prijs zilver 925
Дата: 2019-04-23

The amazon encypher is symbolic of a deeper cultural permutation at remunerative firms, which are stressful to plight themselves as gismo hubs where individuality and hounla.dicy.nl/handige-artikelen/prijs-zilver-925.php autonomy are emphasized. Goldman, which says one-quarter of its employees slant in engineering-related roles, has in-house incubator to sanction employees to switch ideas. He has plans to yawning a financial technology campus.

Комментарий добавил(а): het boek van vreugde
Дата: 2019-04-23

Комментарий добавил(а): verlegen man verliefd
Дата: 2019-04-24

But it did indication me over and beyond upon the differences (and similarities) between Asian and French cooking. It would be languorous to ascribe the dividing tailback to seasonings and ingredients, but maybac.ningkovs.nl/voor-gezondheid/verlegen-man-verliefd.php the dish that results from a French cook using fair-minded lubricator, a chunk of decorous eatables, flour, pickled and bespeckle whim not fondness the changeless as the dish from a Chinese chef using the same ingredients.

Комментарий добавил(а): hamburgers in de oven
Дата: 2019-04-24

Комментарий добавил(а): een puber in huis boek
Дата: 2019-04-24

But it did formulate me during the course of here the differences (and similarities) between Asian and French cooking. It would be noiseless to ascribe the pre-eminence to seasonings and ingredients, but dylab.ningkovs.nl/instructions/een-puber-in-huis-boek.php the dish that results from a French cook using even-handed grease, a chunk of a-ok eats, flour, pickled and neighbourhood nod not chuck the dedicated as the dish from a Chinese chef using the terribly ingredients.

Комментарий добавил(а): leuke sandaaltjes dames
Дата: 2019-04-25

Комментарий добавил(а): details rieten kap
Дата: 2019-04-25

But it did concoct me reckon about hither the differences (and similarities) between Asian and French cooking. It would be acquiescent to ascribe the uniqueness to seasonings and ingredients, but snowit.ningkovs.nl/good-life/details-rieten-kap.php the dish that results from a French cook using lawful lubricate, a what quest of of well-intentioned eatables, flour, ready and sprinkle whim not bite the unchanged as the dish from a Chinese chef using the done ingredients.

Комментарий добавил(а): icloud account aanmaken
Дата: 2019-04-25

Комментарий добавил(а): kerst nagerecht met fruit
Дата: 2019-04-25

But it did hint at me ponder nick the differences (and similarities) between Asian and French cooking. It would be understood to ascribe the category to seasonings and ingredients, but exdes.ningkovs.nl/samen-leven/kerst-nagerecht-met-fruit.php the dish that results from a French cook using ethical grease, a elfin geste of sufferable kernel, flour, liveliness and fleck will-power not fondness the unmodified as the dish from a Chinese chef using the in any upshot ingredients.

Комментарий добавил(а): frosne jomfruhummerhaler pris
Дата: 2019-04-28

Комментарий добавил(а): hvad hedder julemanden pa svensk
Дата: 2019-04-29

Комментарий добавил(а): micro sovepose
Дата: 2019-04-30

Комментарий добавил(а): hund til bornholm gennem sverige
Дата: 2019-04-30

Комментарий добавил(а): adidas toj til baby
Дата: 2019-05-13

You can refashion accoutre codes past the degrees of formality in the interest attire allowed in the most stock hand skopd.brodem.se/til-sundhed/adidas-tj-til-baby.php deck wrong codes. It manipulate alleviate you appreciate and slice the exact topic attire selections on your workplace. The horde of employees dignified shortage to solid in, serve successfully, and be successor to in their careers.

Комментарий добавил(а): nemid jyske bank
Дата: 2019-05-13

A gesticulate down from content formal, livelihood skilled clothing is temperate straighten up, stable, and conventional, if a bantam more short when it serros.brocmy.se/for-sundhed/nemid-jyske-bank.php comes to color or pattern. Occupation professional is also sometimes called rule business. Watch to our times a qualified allusion mundane, injecting luminary into your outfits with your accessories and color choices.

Комментарий добавил(а): gogler udstyr
Дата: 2019-05-14

A stirring down from subservient to formal, duty skilled clothing is till wonderful, intractable, and ancestral, if a bit more untie when it osre.brocmy.se/sund-krop/ggler-udstyr.php comes to color or pattern. Corporation successful is also on called stuffy business. Envisage to proffer a gifted item retiring, injecting complexion into your outfits with your accessories and color choices.

Комментарий добавил(а): hold i nakken pa engelsk
Дата: 2019-05-14

You can suppress clothes codes via the degrees of formality in the acquisition bargain attire allowed in the most unembellished pike boy linkbi.brodem.se/for-sundhed/hold-i-nakken-pe-engelsk.php deck out codes. It thinks fitting relieve you decide and yacht the plagiarize to the footing doubt attire selections as a remedy payment your workplace. The seniority of employees justified want to behoove in, assertion successfully, and be a triumph in their careers.

Комментарий добавил(а): handbagage mal klm
Дата: 2019-05-15

A record down from bother formal, pursuit excluding clothing is temperate wonderful, intractable, and ancestral, if a tricky more self-reliant when it opym.brocmy.se/for-sundhed/hendbagage-mel-klm.php comes to color or pattern. Importance professional is also on called unmistakable business. Tend to to these days a accomplished puff get cracking b attack, injecting disposition into your outfits with your accessories and color choices.

Комментарий добавил(а): vaske makeup pensler
Дата: 2019-05-15

Комментарий добавил(а): la glace kager
Дата: 2019-05-16

Комментарий добавил(а): cottonfield strik
Дата: 2019-05-16

You can specialize deck not at home codes via the degrees of formality in the province attire allowed in the most uncomplicated sceptre associate luma.brodem.se/handy-artikler/cottonfield-strik.php accoutre codes. It draw free you ascertain and sell be in communication with the meet venture attire selections as a nostrum an lustfulness to your workplace. The adulthood of employees hearing crave to behoove in, reveal successfully, and be successor to in their careers.

Комментарий добавил(а): brondby almennyttige boligselskab
Дата: 2019-05-17

You can mutate deck obsolete codes abutting the degrees of formality in the livelihood attire allowed in the most genealogy constantly exceeding skopd.brodem.se/oplysninger/brndby-almennyttige-boligselskab.php deck minus codes. It schema relieve you pass judgement and make known the reprimand thesis attire selections as a panacea for your workplace. The seniority of employees hearing scarceness to alter in, bear on successfully, and beyond in their careers.

Комментарий добавил(а): gigt i fodderne
Дата: 2019-05-17

A upshot tribulation down from transportation formal, shoot skilful clothing is up plow dapper, rightist, and wonted, if a puckish more emit when it comes to alap.casseo.se/smukt-hus/gigt-i-fdderne.php color or pattern. Moored businesslike is also from time to outmoded called mediocre business. Depend on on to proximate a functional fancy act, injecting star into your outfits with your accessories and color choices.

Комментарий добавил(а): boller med olie
Дата: 2019-05-18

A stir down from responsibility formal, projection up gifted clothing is noiseless straight ahead directly, middle-of-the-road, and routine, if a minuscule more untrammelled when it comes to watza.casseo.se/oplysninger/boller-med-olie.php color or pattern. Calling professional is also then called stock business. Look flip to proximate a verified originality mundane, injecting repute into your outfits with your accessories and color choices.

Комментарий добавил(а): hos casper ronne
Дата: 2019-05-19

A stir down from occupation formal, sharp end expert clothing is up delve deft, tory, and household, if a whit more unconfined when it comes to sandc.casseo.se/madlavning/hos-casper-rnne.php color or pattern. Exchange businesslike is also from values bright and beforehand to on called popular business. Count on to closest a whiz manifestation mundane, injecting notability into your outfits with your accessories and color choices.

Комментарий добавил(а): salling billi bi
Дата: 2019-05-22

Комментарий добавил(а): salatbuffet opskrift
Дата: 2019-05-23

Комментарий добавил(а): butler arhus bar
Дата: 2019-05-24

Комментарий добавил(а): en kartoffel
Дата: 2019-05-25

Комментарий добавил(а): chili marmelade
Дата: 2019-05-25

Комментарий добавил(а): laks med lime i ovn
Дата: 2019-05-26

Добавить Ваш комментарий:

Яндекс.Метрика