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Home Question Bank Math Geometry Archive 2024 December 29

Geometry Questions from Dec 29,2024

Browse the Geometry Q&A Archive for Dec 29,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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ما هو الشكل الناتج عن نشر مكعب؟ 10 For positive real numbers \( a \), and \( b \), the lines \( l_{1} \) and \( l_{2} \) have equations given by \[ l_{1}: \mathbf{r}=\left(\begin{array}{l} 1 \\ 2 \\ 5 \end{array}\right)+\lambda\left(\begin{array}{c} -1 \\ 0 \\ 3 \end{array}\right) \text { for } \lambda \in \mathbb{R} \text { and } l_{2}: \mathbf{r}=\left(\begin{array}{l} a \\ 9 \\ 9 \end{array}\right)+\mu\left(\begin{array}{l} 1 \\ 7 \\ b \end{array}\right) \text { for } \mu \in \mathbb{R} \] It is given that the two lines intersect each other. (i) Show that \( 3 a=b+2 \). [2] (ii) Find the value of \( b \) if the angle between the two lines is \( \cos ^{-1}\left(\frac{11}{\sqrt{660}}\right) \) radians. The equation of the plane \( \Pi_{1} \) that contains both lines is now given by \( \mathbf{r} \cdot\left(\begin{array}{c}3 \\ -1 \\ 1\end{array}\right)=6 \). (iii) A second plane \( \Pi_{2} \) meets \( \Pi_{1} \) in the line \( l_{1} \) and contains the point \( (2,1,3) \). Find an equation for \( \Pi_{2} \) in scalar product form. [2] (iv) Hence find the angle between the planes \( \Pi_{1} \) and \( \Pi_{2} \). [2] (v) The plane \( \Pi_{3} \) is parallel to the plane \( \Pi_{1} \). Given that the distance between both planes is \( \frac{7}{\sqrt{11}} \) units and that \( \Pi_{3} \) is closer to the origin than \( \Pi_{1} \), find a cartesian equation for \( \Pi_{3} \). [3] \begin{tabular}{l|l} \hline 10 & (b) 4 (c) \( r \cdot\left(\begin{array}{l}3 \\ 1 \\ 1\end{array}\right)=10 \$ \). \end{tabular} (d) \( 0.613 \mathrm{rad}(3 \mathrm{sf}) \) or \( 35.1^{\circ}\left(\right. \) nearest \( \left.0.1^{\circ}\right) \) (e) \( 3 x-y+z=-1 \) إذا كان لديك صندوق يتمتع بحجم ثابت قدره 1000 سم³، احسب الأبعاد التي ستؤدي إلى الحد الأدنى من المساحة السطحية. Si un cilindro tiene una altura que es el doble del radio, y el volumen del cilindro es \( 72\pi \) cm³, determina el valor del radio. John painted his most famous work, in his country, in 1930 on composition board with perimeter 103.59 in. If the rectangular painting is 5.65 in. taller than it is wide, find the dimensions of the painting. What is the width? 23.07 in. (Simplify your answer. Type an integer or decimal rounded to two decimal places as needed.) What is the length? (Simplify your answer. Type an integer or decimal rounded to two decimal places as needed.) 23. Spatis When Alonzo, who is \( 5^{\prime} 11^{\prime \prime} \) tall, stands next to a basketball goal, his shadow is \( 2^{\prime} \) 'ong, and the basketball goal's shadow is \( 4^{\prime} 4^{\prime \prime} \) long. About how tall is the basketbail goal? Un círculo tiene un centro. ¿Dónde se encuentra este centro en el círculo? Soit ABC un triangle et I le milieu de \( [\mathrm{BC}] \). H est le point tel que \( \overrightarrow{\mathrm{AH}}=\frac{3}{4} \overrightarrow{\mathrm{AI}} \). 1) construire les points E , le projeté de H sur \( (\mathrm{BC}) \) parallèlement à \( (\mathrm{AB}) \), et F le projeté de \( H \) sur ( BC\( ) \) parallèlement à \( (\mathrm{AC}) \) 2) Montrer que : \( \overrightarrow{\mathrm{BE}}=\frac{3}{4} \overrightarrow{\mathrm{BI}} \) et \( \overrightarrow{\mathrm{CF}}=\frac{3}{4} \overrightarrow{\mathrm{Cl}} \) 3) En déduire que I est le milieu de \( [\mathrm{EF}] \) In the opposite figure \( \overrightarrow{\mathrm{DC}} \) bisects \( \angle \mathrm{ACB} \) , \( \mathrm{AE}=12 \mathrm{~cm} \). , \( \mathrm{CE}=6 \mathrm{~cm} \). \( \mathrm{BC}=9 \mathrm{~cm} \). rove that \( : \overline{\mathrm{DE}} / / \overline{\mathrm{BC}} \) Estnerzio 13.24 Trovare, se esiste, un'applicazione lineare \( F: \mathbb{R}^{4} \longrightarrow \mathbb{R}^{4} \) che trasfor- ma i vettori \( (1,-1,-1,0) \) e \( (0,0,0,1) \) in sé stessi e che trasforma i vettori \( (1,0,0,-1) \) e \( (0,1,-1,0) \) nei loro opposti. \( \mathbf{9 6} \) ESERCIZI DI GEOMETAIA
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