Državno natjecanje 2001 SŠ4 2
Dodao/la:
arhiva1. travnja 2012. Papir oblika kvadrata s vrhovima
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
i
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
ima stranica duljina
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
. Na njegovim stranicama
![\overline{FB}](/media/m/d/d/6/dd612921cd0736e37f78edb04ff86a63.png)
i
![\overline{BH}](/media/m/a/0/a/a0a08d5cdcd9a19121ed4901ee25a875.png)
označenje su točke
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
i
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, odnosno
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
i
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
, takve da je
![|FG| = |GA| = |AB|](/media/m/2/4/d/24d4e053f40adbf5b02a401f2ce327fb.png)
i
![|BE| = |EC| = |CH|](/media/m/b/5/0/b50850ed6fa3b5a04423695860626e6b.png)
. Papir je presavinut po dužinama
![\overline{DG}](/media/m/f/0/d/f0d406a116ca8492421ed4cc23a8c306.png)
,
![\overline{DA}](/media/m/8/4/5/845d2fffb3c5eb6412b26a001c3b4b4d.png)
,
![\overline{DC}](/media/m/d/4/8/d4833b07ff9ba6f723009f06316626fd.png)
i
![\overline{AC}](/media/m/d/9/5/d95354f0f833a5fda9c16a01a878c14f.png)
tako da se točka
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
poklopi s
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
, a točke
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
i
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
s točkom
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
. Odredite volumen tako nastale trostrane piramide
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
.
%V0
Papir oblika kvadrata s vrhovima $F$, $B$, $H$ i $D$ ima stranica duljina $a$. Na njegovim stranicama $\overline{FB}$ i $\overline{BH}$ označenje su točke $G$ i $A$, odnosno $E$ i $C$, takve da je $|FG| = |GA| = |AB|$ i $|BE| = |EC| = |CH|$. Papir je presavinut po dužinama $\overline{DG}$, $\overline{DA}$, $\overline{DC}$ i $\overline{AC}$ tako da se točka $G$ poklopi s $B$, a točke $F$ i $H$ s točkom $E$. Odredite volumen tako nastale trostrane piramide $ABCD$.
Izvor: Državno natjecanje iz matematike 2001