📐 geometry

1. **Problem statement:** Given a circle with points A, B, C, D, E on the circumference and center O, with EOB a straight line, \(\angle AÔB = 105^\circ\) and \(\angle BÊC = 20^\ci

1. **Problem statement:** Given \(\angle ABP = 76^\circ\), find the value of \(y\) in the figure. 2. **Relevant concept:** In circle geometry, the angle formed outside the circle b

1. The problem is to analyze the given figure and solve the first question related to it. 2. Since the figure is not provided in text, please describe the problem or provide the fi

1. **Problem statement:** We need to find how many triangles are formed in a trapezoid divided into four horizontal layers by three parallel lines, with one diagonal from the top-l

1. **Stating the problem:** We need to find how many triangles are in the given geometric figure, which is a large triangle subdivided by three segments connecting vertices to poin

1. **Problem statement:** We have a table with a circular top and a circular base.

1. **Problem statement:** We have a solid cut into two identical halves along the plane of symmetry VAFHCV. We need to calculate the surface area of one of these halves. 2. **Under

1. **State the problem:** Calculate the volume of a solid consisting of a rectangular block with a square base of side length 6 cm and height 10 cm, topped by a right pyramid with

1. **Stating the problem:** We have an obtuse angle $x\hat{O}y$ with vertex $O$. The bisector of this angle is the semi-line $[Ou]$. Points $F$ and $R$ lie on $[OY]$ and $[OX]$ res

1. **State the problem:** We will list and explain all 9 circle theorems, which are fundamental properties relating angles, chords, tangents, and segments in a circle. 2. **Theorem

1. The problem is to understand and list all the main circle theorems. 2. Circle theorems are rules about angles, lengths, and arcs in circles that help solve geometry problems.

1. **Problem statement:** Given a circle with center $O$, a tangent line $BCD$ touching the circle at point $C$, and a point $A$ on the circle, prove that $\angle BAC + \angle ACD

1. The problem is to find the formula for the area of a circle. 2. The area of a circle is the amount of space enclosed within its boundary.

1. **Постановка задачи:** Даны секущие EA и EB, пересекающие окружность с центром в точке O, и хорда CD. Известно, что $AB=AE$ и $BE=12$. Нужно найти длину хорды $CD$. 2. **Формула

1. **State the problem:** We need to find the length of the hypotenuse $QR$ in a right triangle $PQR$ where the right angle is at $P$. The sides adjacent to the right angle are $PQ

1. **Problem statement:** We have a right-angled triangle XYZ with the right angle at Y. Side XY measures 5 cm, side XZ (the hypotenuse) measures 19 cm, and we need to find the len

1. The problem states the Pythagorean Theorem: $$a^2 + b^2 = c^2$$ and gives values $a=7$ and $b=11$. We need to find $c^2$. 2. First, calculate $a^2$ and $b^2$:

1. **Problem statement:** Prove that the straight line joining the origin to the intersection of the line $kx + hy = 2hk$ and the circle $(x - h)^2 + (y - k)^2 = a^2$ is perpendicu

1. The problem involves understanding the dimensions of a geometric shape with given lengths: total width 210 units, height 170 units, a 65-unit segment from the left to a notch, a

1. **Stating the problem:** We have two pieces, each a rectangle of dimensions 6 cm by 2 cm, and we want to arrange them to form a figure with a perimeter of 22 cm. 2. **Understand

1. **Problem statement:** A solid toy is made by mounting a cone on top of a hemisphere. The height of the cone is 7 cm and the radius of both the cone and hemisphere is 3.5 cm. We