📐 geometry

1. **Problem Statement:** We have a circle with center $O$, a tangent line $Pi$ touching the circle at point $i$, and a chord $\overline{iQ}$.

1. Bài toán yêu cầu cho đường tròn tâm $O$ bán kính $R$ với đường kính $AB$. 2. Điểm $C$ là trung điểm của đoạn $OB$, tức là $C$ nằm trên đoạn thẳng nối $O$ và $B$ sao cho $OC = CB

1. **Problem Statement:** Construct quadrilateral ABCD with given sides $AB=4.5$ cm, $AC=CD=5$ cm, $AD=6$ cm, and angle $\angle BAC=60^\circ$. Then construct triangle PBC such that

1. **Problem statement:** Find the volume of the tetrahedron bounded by the coordinate planes $x=0$, $y=0$, $z=0$ and the plane $$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1.$$\n\n

1. **Stating the problem:** We have a triangle ABC with a point D on BC such that AD is perpendicular to BC. The angle at A adjacent to this perpendicular is 40°. At vertex B, ther

1. **State the problem:** We need to find the measure of angle $C$ in a triangle where sides $a=18$, $b=22$, and $c=20$ are given. 2. **Formula used:** The Law of Cosines states:

1. **Problem statement:** An aquarium is in the form of a cuboid with external dimensions 80 cm x 80 cm x 40 cm. We need to find the area of paper required to cover the bottom, sid

1. **Problem:** The capacity of a closed cylindrical vessel of height 1 m is 15.4 L. Find the surface area of metal sheet needed to make it. 2. **Given:**

1. **State the problem:** We are given a rectangle where the length is 20% more than the breadth, and the breadth is 20 cm. We need to find the area of the rectangle. 2. **Identify

1. **Problem Statement:** We have two parallelograms ABCD and AFGD. AF and CD intersect at point E. Given that $AD = AE$ and $\angle DGF = 36^\circ$, we need to find $\angle BCD$.

1. **Problem Statement:** Find the volume of water left in a right circular cylinder of radius 60 cm and height 180 cm after placing a solid consisting of a right circular cone of

1. **State the problem:** Given a circle with radius 7 cm and a sector with central angle 150°, we want to find the length of the chord (the other side) opposite the 150° angle. 2.

1. **State the problem:** We need to find the area of the blue shaded region in a circle of radius 7 cm, where the sector has a central angle of 150°. The blue region is the sector

1. **Problem Statement:** Find the coordinates of points H and G, the midpoint of segment MQ, and the distance between points A and B or the area of rectangle RQFG based on the giv

1. **State the problem:** Given the perimeter of the rhombus VWXY is 20.5 cm, find the length of one side. 2. **Recall the formula for the perimeter of a rhombus:**

1. **Stating the problem:** We are given two angles: one is $130^\circ$ and the other is $(28 + 10)^\circ$. We need to find the value of $(2a - b)^\circ$ based on these angles.

1. **Problem Statement:** We need to find the measure of the exterior angle $q$ at the top vertex of the triangle. 2. **Key Concept:** The exterior angle of a triangle is equal to

1. The problem is to solve various problems related to circles (circulus). 2. The main formulas used in circle problems include:

1. The problem is to understand the term "circulus" which is Latin for "circle". 2. A circle is a set of points in a plane that are all at the same distance (radius) from a fixed p

1. **Problem statement:** We have a right triangle with vertices C, B, and D, where the right angle is at vertex B. The leg BC adjacent to the right angle is 7 miles long. We need

1. **Problem:** Find the equation of the plane through the points (3, 0, -1), (-2, -2, 3), and (7, 1, -4). 2. **Formula and rules:** The equation of a plane through three points $P