Question

7. Se consideră cubul ABCDEFGHI din Figura 3, cu muchia de 6 cm. a) Calculează lungimea diagonalei cubului. b) Determină distanta de la punctul C la diagonala BH. c) Determină distanta de la punctul A la planul (BCG). d) Determină distanta de la punctul E la planul (DBF).​

Answer 1

a.

AB=6cm

Diagonala unui cub=l√3

Deci BH=AG=6√3cm

b.

HC=l√2 (diagonala patrat)

HC=6√2cm

BH²=HC²+BC²

108=72+36

108=108⇒ R.T.P⇒ ΔBHC dr in C

$d(C,BH)=\frac{CH\times BC}{BH} \ (formula\ inaltimii\ in\ triunghiul\ dreptunghic)\\\\d(C,BH)=\frac{6\sqrt{2}\times 6 }{6\sqrt{3} } =\frac{6\sqrt{6} }{3} =2\sqrt{6} \ cm$

c.

AB⊥BC

CM⊥BG⇒ d(A,(BCG)=AM

• In ΔAMB dr in M aplicam Pitagora

AM²=AB²+BM²

BM=6√2:2=3√2 cm

AM²=36+18

AM²=54

AM=3√6 cm

d.

DF=6√3 cm(diagonala cub)

ΔDFB dr in B

Fie BN⊥FD

BN=2√6 cm=d(C,BH)

EF⊥BF

BN⊥FD⇒ d(E,(BDF))=EN

EF²=36

DE²=72

DF²=108⇒DF²=EF²+DE²⇒ R.T.P ⇒ΔEDF dr in E⇒ EN inaltime in triunghi

$EN=\frac{ED\times EF}{FD} =\frac{6\sqrt{2} \times 6}{6\sqrt{3} } =2\sqrt{6}\ cm$