📐 geometry

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

1. **نبدأ بملاحظة المعطيات:** لدينا نقطة $P(-1,0,1)$ ومستقيم معطى بالنسبة للمعادلات: $$\frac{1 - y}{2} = \frac{1 - y}{1} = \frac{6 + 1}{1 - z}$$

**Problem 39:** Show that the triangle with vertices A(0, 2), B(-3, -1), and C(-4, 3) is isosceles. 1. Calculate the lengths of the sides AB, BC, and CA using the distance formula:

1. **Problem Statement:** (i) Show that points A(0, 2), B(\sqrt{3}, -1), and C(0, -2) form a right-angled triangle.

1. **Problem (i):** Show that A(0,2), B(\(\sqrt{3}\),-1), C(0,-2) form a right-angled triangle. 2. Calculate lengths of sides using distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^

1. প্রশ্ন বোঝা: একটি ধাতব গোলকের ব্যাসের ক্ষেত্রফল $A$ থেকে নতুন গোলকের ব্যাসের ক্ষেত্রফল $\frac{A}{2}$ হয়েছে। এর অর্থ, নতুন গোলকের ব্যাসের ক্ষেত্রফল আগের গোলকের ক্ষেত্রফলের অর্ধেক

1. Given a right triangle ABC with \(\angle B = 90^\circ\), \(\angle C = 60^\circ\), and hypotenuse \(AC = 10\) cm. a) Find \(\angle A\).