📐 geometry

1. **Problem statement:** We have a cylinder with diameter 7 cm and height 9 cm.

1. O problema envolve calcular volumes de uma base esférica e de um cilindro, e depois determinar a altura do creme no cilindro quando o volume é metade do total. 2. Fórmulas usada

1. مسئله: در مثلث ABC، پاره‌خط DE موازی BC است و D روی AB و E روی AC قرار دارند. باید مقادیر $x$ و $y$ را پیدا کنیم. 2. قانون استفاده شده: اگر دو خط موازی باشند، نسبت‌های طول‌های ب

1. **Stating the problem:** We have a circle (C) with center O and diameter BE of length 6 cm. Point A lies on the extension of line BE such that OA = 5 cm.

1. **Determine the angle \(\theta\) at which the tangent line is drawn.** Given the tangent line at point A and chord AB, the angle between the tangent and chord equals the measure

1. **Problem statement:** We have a pentagon with exterior angles marked. The sum of all but one of these exterior angles is given as $\frac{23\pi}{6}$ radians. We need to find the

1. Planteamiento del problema: Dados los segmentos AE = 16, CD = 16, CE = 12, se pide encontrar EB. 2. Observamos que AE y EB son segmentos adyacentes sobre una línea, por lo que A

1. **State the problem:** Find the surface area of a rectangular prism with dimensions length $= 12$ cm, width $= 15$ cm, and height $= 3$ cm. 2. **Formula for surface area of a re

1. **State the problem:** We need to find the surface area of a cube with each edge measuring 7 cm. 2. **Formula for surface area of a cube:** The surface area (SA) of a cube is gi

1. **State the problem:** Find the surface area of a rectangular prism with dimensions 6 cm, 8 cm, and 12 cm. 2. **Formula for surface area of a rectangular prism:**

1. **State the problem:** We have three points A, B, and C on a circle with angles labeled as $x$, $2(2x + 20)$ degrees, and $5x + 40$ degrees. We need to find the value of $x$, th

1. **State the problem:** We are given trapezium PQRS with angles at S and R expressed as $(3x - 24)^\circ$ and $(102 - x)^\circ$ respectively. Angles PQR and QPS are right angles.

1. **Problem Statement:** Identify three pairs of lines that are parallel, three pairs that are perpendicular, and three pairs that intersect but are not perpendicular. Then find t

1. The problem asks to name points, line segments, rays, and angles based on the given figure with lines a and b intersecting at point X. 2. Three points on line a are: P, X, Q.

1. **Problem:** Calculate the exact area of an equilateral triangle with each side length 24 cm, which is used as a triangular bracket glued to the back of a circular mirror of rad

1. **Problem Statement:** Given a circle with center $C$, chord $MP$ with midpoint $E$, diameter $F$, and angle $\angle HCP = 130^\circ$, find the angles $\angle HMP$, $\angle HEM$

1. **Problem:** Herr Schmidt möchte für sein rundes Schwimmbecken mit Durchmesser $d=5,0$ m eine Abdeckplane kaufen, die ringsum 0,5 m über den Beckenrand hängt. 2. **Gesucht:** Di

1. The problem is to understand the concept of rotation in geometry. 2. Rotation is a transformation that turns a figure about a fixed point called the center of rotation.

1. The problem asks to identify which shapes are rotations of shape X. 2. Shape X consists of 4 squares: a horizontal row of 3 squares with 1 extra square attached below the leftmo

1. The problem involves constructing a congruent angle at point R to the angle at Q using compass and straightedge. 2. The first step in copying an angle is to draw an arc centered

1. **Problem statement:** Find the magnitudes (modules) of vectors \(\overline{AD}\), \(\overline{AC}\), and \(\overline{AC_1}\) in a cube \(ABCDA_1B_1C_1D_1\) with edge length 1.