Explain the relationship between the area of cross-sections of a three-dimensional shape and its total volume.

\( A B C \) est un triangle et \( I \) milieu de \( [B C] \). 1) Soit \( \operatorname{Ebar}\{(A, 1) ;(B, 2) \),\( \} et F b a r\{(B, 2) ;,(C, 1)\} \). a) Montrer que: \( \overrightarrow{A E}=\frac{2}{3} \overrightarrow{A B} \). b) Montrer que: \( \overrightarrow{B F}=\frac{1}{3} \overrightarrow{B C} \) c) Construire les points \( I, E \) et \( F \). 2) Soit \( G b a r\{(A, 1) ;(B, 2) ;,(C, 1)\} \). a) Montrer que \( \operatorname{Gbar}\{(A, 1) ;(F, 3) \),\( \} . \) b) Montrer que \( G \in(E C) \). c) Montrer que \( G \in(I B) \) et construire \( G \). 3) Determiner l'ensemble des points \( M \) du plan qui verifient chacune des équations suivantes: a) \( \|\overrightarrow{M A}+2 \overrightarrow{M B}+\overrightarrow{M C}\|=8 \) b) \( \|\overrightarrow{M A}+2 \overrightarrow{M B}\|=\|\overrightarrow{M C}+2 \overrightarrow{M B}\| \)

A square prism measuring 2 m by 2 m in base length and 5 m in height has a half-sphere attached to the top. What is the total surface area of this composite solid?

the area of \( \triangle A B C=12 \mathrm{~cm}^{2} \), then \( \left(b^{2}+c^{2}-a^{2}\right) \tan A=\cdots \) 12 (b) 24

Dado un cilindro con un radio de 4 cm, ¿cuál debe ser la altura para que el volumen sea \( 64\pi \) cm³?

3) The exterior bisector of the vertex angle of an isosceles triangle \( \cdots \cdots . . . \). the base. \( \begin{array}{ll}\text { (a) bisects } & \text { (b) perpendicular to } \\ \text { (c) intersect } & \text { (d) parallel }\end{array} \)