Given the equations of lines \( \boldsymbol{l}_{1} \) and \( \boldsymbol{l}_{2} \) are \( \mathrm{r}=(\mathrm{i}+2 \mathrm{j}+3 \mathrm{k})+\lambda(\mathrm{i}-\mathrm{j}+2 \mathrm{k}) \) and \( \mathrm{r}=(5 \mathrm{i}+\mathrm{j}+5 \mathrm{k})+\mu 2 \mathrm{i}+\mathrm{j}-2 \mathrm{k}) \) respectively, where \( \lambda \) and \( \mu \) are real patameters. (a) Show that lines \( \boldsymbol{\ell}_{1} \) and \( \boldsymbol{\ell}_{2} \) intersect and hence, state the coordinates of the point of intersection. (b) Find the Cartesian equation of the plane \( \Pi_{1} \), which contains lines \( \boldsymbol{\ell}_{1} \) and \( \boldsymbol{\ell}_{2} \). (c) The planes \( \Pi_{2} \) and \( \Pi_{3} \) are with equations \( 2 \mathrm{r}+2 \mathrm{y}+z=3 \) and \( 2 \mathrm{r}+p y+2 z=2 \) respectively, where \( p \) is a constant. (i) Find the vector equation of the line of intersection between planes \( \Pi_{1} \) and \( \Pi_{2} \). (ii) Determine the value of \( p \) such that the planes \( \Pi_{1}, \Pi_{2} \) and \( \Pi_{3} \) do not have any common point.

elect all of the options which are true of the perpendicular bisector of line AB

Soit MNP un triangle tels que : \( \mathrm{MN}=6 \mathrm{~cm} ; \mathrm{NP}=4 \mathrm{~cm} \) et \( \mathrm{MP}=7 \mathrm{~cm} \). Construire les médianes de ce triangle

Exercice 2 Soit \( A B C \) in triangle. Soit D et E les points vérifiant : \( \overrightarrow{A D}=\frac{2}{3} \overrightarrow{A B} \) et \( \overrightarrow{A E}=\frac{2}{5} \overrightarrow{A C} \). F est le point d'intersection de \( (A B) \) avec la parallèle à \( (B C) \) passant par E. G est le point d'intersection des droites \( (B C) \) et (ED). 1. Exprimer \( \overrightarrow{A F} \) en fonction de \( \overrightarrow{A B} \). 2. Exprimer \( \overrightarrow{D B} \) en fonction de \( \overrightarrow{A B} \), puis \( \overrightarrow{D F} \) en fonction de \( \overrightarrow{A B} \). 3. En déduire \( \overrightarrow{D F} \) en fonction de \( \overrightarrow{D B} \). 4. Exprimer en fonction de \( \overrightarrow{A B} \) et de \( \overrightarrow{A C} \) les vec- teurs \( \overrightarrow{D E}, \overrightarrow{D C} \).

Un triángulo rectángulo tiene un cateto que mide 6 unidades y la hipotenusa mide 10 unidades. ¿Cuál es la longitud del otro cateto?

Точка \( D \) делит основание \( A B \) равнобедренного треугольника \( A B C \) в отношении \( 2: 1 \), считая от вершины \( A \). Радиус окружности, описанной около треугольника \( A D C \), равен 12 . Найдите радиус окружности, описанной около треугольника \( B D C \).