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Home Question Bank Math Geometry Archive 2025 January 10

Geometry Questions from Jan 10,2025

Browse the Geometry Q&A Archive for Jan 10,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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Given: \( D \) is the midpoint of \( \overline{A C}, \overline{B A} \cong \overline{B C} \) and \( \angle E D A \cong \angle F D C \). Prove: \( \triangle A E D \cong \triangle C F D \). Note: quadrilateral properties are not permitted in this proof. (2) If \( \mathrm{m}(\angle \mathrm{B})=80^{\circ} \), then \( \mathrm{m}( \) refle \( x \angle B)=\ldots \) \( \begin{array}{llll}\text { (a) } 10^{\circ} & \text { (b) } 100^{\circ} & \text { (c) } 80^{\circ} & \text { (d) } 280^{\circ}\end{array} \) What is the formula for the surface area of a sphere in terms of its radius? Esercizio 1. Si consideri la conica \[ \mathscr{C}: 3 x^{2}-10 x y+3 y^{2}+4 x-2 y-1=0 \] (a) Provare che si tratta di una iperbole generale. (b) Calcolarne centro e asintoti. Esercizio 3. Trovare centro e raggio della circonferenza \( \mathscr{C}:\left\{\begin{array}{l}x^{2}+y^{2}+z^{2}-4 x+2 y-6 z=1 \\ x+2 y-2 z=0\end{array}\right. \) Esercizio 5. Si consideri la conica proiettiva: \[ \mathscr{C}: X_{1}^{2}+2 X_{1} X_{2}+X_{2}^{2}+3 X_{0}^{2}+2 X_{1} X_{0}=0 \] (a) Trovare la forma canonica di \( \mathscr{C} \) (b) Trovare i punti impropri di \( \mathscr{C} \) (c) Calcolare rango e tipo di \( \mathscr{C} \). Esercizio 5. Si consideri la conica proiettiva: \[ \mathscr{C}: X_{1}^{2}+2 X_{1} X_{2}+X_{2}^{2}+3 X_{0}^{2}+2 X_{1} X_{0}=0 \] (a) Trovare la forma canonica di \( \mathscr{C} \). (b) Trovare i punti impropri di \( \mathscr{C} \). (c) Calcolare rango e tipo di \( \mathscr{C} \). Esercizio 5. Si consideri la conica proiettiva: \[ \mathscr{C}: X_{1}^{2}+2 X_{1} X_{2}+X_{2}^{2}+3 X_{0}^{2}+2 X_{1} X_{0}=0 \] (a) Trovare la forma canonica di \( \mathscr{C} \) (b) Trovare i punti improni di \( \mathscr{C} \). Esercizio 2. Portare in forma canonica la quadrica \[ \mathscr{Q}: x^{2}-2 x y+2 y^{2}+2 y z+z^{2}+2 x+2=0 \] Esercizio 2. Portare in forma canonica la quadrica \[ \mathscr{Q}: x^{2}-2 x y+2 y^{2}+2 y z+z^{2}+2 x+2=0 \]
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