📐 geometry

1. Let's consider the problem about triangles. Since no specific problem is given, we'll review the general properties of triangles and how to find the area. 2. A triangle is a pol

1. **Problem statement:** We are given points $A(1,1,0)$, $B(2,0,0)$, $C(1,3,-1)$, and $E(2,2,2)$. **a. Find the equation of plane $(P)$ determined by points $A$, $B$, and $C$.**

1. **Problem statement:** Find the volume of a triangular prism with base sides 3 cm, 4 cm, and 5 cm, and height 10 cm. 2. The triangle with sides 3, 4, and 5 is a right triangle (

1. **State the problem**: We need to find the three-figure bearings from the centre to Town A and Town C based on given angles. 2. **Understand bearings**: Bearings are measured cl

1. Problem statement: We want to find the shortest length of a bridge connecting two islands. The first island is shaped like a parabola with height 4 km, and the second is a circl

1. State the problem: Find the obtuse angle that the line segment AB forms with the x-axis, where A = $(4,-3)$ and B = $(8,5)$.\n2. Calculate the slope of line AB using the formula

1. **Problem statement:** Show that the Cartesian plane is a model of incidence geometry, i.e., it satisfies the axioms of incidence geometry. 2. **Axiom 1: Any two distinct points

1. Solve for x in the equation provided (equation missing, please provide specific equations for precise solution). 2. Solve for x in the equation provided (equation missing, pleas

1. **State the problem:** We need to prove that the perpendicular bisector of a chord in a circle bisects the central angle subtended by the chord. 2. **Set up the scenario:** Cons

1. The problem describes a composite geometric figure consisting of several connected polygons: a rectangle, right triangles, and triangular indentations. 2. To analyze this figure

1. Problem: Find the distance between the points $A(3,4)$ and $B(-1,1)$.\n\n2. The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the Cartesian plane is given by th

1. **State the problem:** Find the length of the arc for each sector given radius and central angle. The formula for arc length $L$ is:

1. The problem is to find the length of an arc for each given radius and central angle. 2. The formula for arc length is given by $$L = 2\pi r \cdot \left(\frac{\theta}{360}\right)

1. **Understanding the problem:** We have a triangle with a horizontal base labeled $z$, a left side at a $30^\circ$ angle with the base with hypotenuse length $220$ ft, a vertical

1. The problem asks us to find the volume of a beach ball which is a sphere. 2. We are given the diameter of the sphere as 30 cm. To find the radius $r$, we use the relation:

1. **State the problem:** We need to find the volume of a sphere given its diameter is 16 mm. The formula for the volume of a sphere is $$V = \frac{4}{3} \pi r^3$$ where $r$ is the

1. The problem states that the volume $V$ of a sphere is given by the formula $$V=\frac{4}{3}\pi r^3$$ where $r$ is the radius of the sphere. 2. We are given that the radius $r=13\

1. The problem states that we need to find the volume of a sphere with radius $r = 9$ m. 2. The formula for the volume of a sphere is given by:

1. Énoncé du problème : Trouver l'équation de la troisième route passant par le point d'intersection des deux premières routes et perpendiculaire à la route auxiliaire. 2. Trouvons

1. Stating the problem: We have a triangle ABC where the sides satisfy the relation $10a = 12c = 13b$. We need to find the smallest angle in the triangle. 2. Express the sides in t

1. **State the problem**: We want to find the area of a trapezium (trapezoid), which is a quadrilateral with exactly one pair of parallel sides. 2. **Identify the sides**: Let the