📐 geometry

1. **Problem statement:** We have four concentric circles centered at point $M$ with points $N, P, R, S, T, U, V, Y$ on the circles. The segments $|MN| = |NP| = |PR| = |RS|$ are eq

1. **State the problem:** We need to find the volume of a pentagonal prism with a height of 13 cm, each side of the pentagonal base measuring 13 cm, and the apothem (distance from

1. **State the problem:** We need to find the height $H$ of the building given a right triangle formed by the building, the ground, and the lamp. 2. **Identify the known values:**

1. The problem involves understanding a geometric figure where AC and BD are not straight lines, and tangents are involved. 2. Since the answer is given as 55, we need to verify or

1. **Stating the problem:** Given that \(\angle AOB = 125^\circ\), find the measure of \(\angle COD\) in the figure where points A, B, C, D, and O form a pentagon with a circle cen

1. **Problem Statement:** We have two roads: the first road (I. road) is horizontal, and the second road (II. road) is diagonal. Point A lies on the first road, and point B lies on

1. **Problem statement:** We need to find the perimeter of a sector of a circle with radius $10.5$ cm and central angle $\frac{2\pi}{3}$. Given $\pi = \frac{22}{7}$. 2. **Formula f

1. **Problem Statement:** Find the maximum number of hemispheres that can be scooped from a cube. 2. **Understanding the problem:** We want to fit hemispheres inside a cube such th

1. **Problem statement:** From one face of a wooden cube with side length 14 cm, hemispheres of diameter 1.4 cm are scooped out. We need to find the maximum number of hemispheres t

1. **Problem statement:** We have a solid right circular cylinder with height $12$ cm and base radius $5$ cm. A right circular cone with the same height and base radius is removed

1. The problem involves identifying and understanding points on the Cartesian plane, including their coordinates and quadrant positions. 2. The Cartesian plane consists of two perp

1. **Problem Statement:** Given parallelogram ABCD, point P lies on AD such that $AP=\frac{1}{3}AD$, and point Q lies on BC such that $CQ=\frac{1}{3}BC$. We need to analyze or solv

1. **Problem statement:** We have a quadrilateral ABCD with AB \parallel DC and BC \parallel AD. Given angles: \angle B = 121^\circ, \angle A = 40^\circ + z^\circ, \angle C = y^\ci

1. The problem asks for the scale factor (as a ratio) of Figure B to Figure A based on the side lengths of two triangles. 2. The scale factor between two similar figures is the rat

1. The problem states that triangles $\triangle GHJ$ and $\triangle LMK$ are similar with a scale factor of $\frac{5}{6}$ from $\triangle LMK$ to $\triangle GHJ$. We are asked to f

1. **Problem Statement:** We have triangle $\triangle PQR$ with circumcenter $V$ where the perpendicular bisectors $SV$, $TV$, and $UV$ intersect. Given: $UV=30$, $PS=64$, and $RV=

1. **Problem Statement:** We have triangle $\triangle XYZ$ with circumcenter $S$, where $S$ is the intersection of the perpendicular bisectors $\overline{TS}$, $\overline{US}$, and

1. **Problem Statement:** We have triangle $\triangle TUV$ with circumcenter $D$, where $D$ is the intersection of the perpendicular bisectors $AD$, $BD$, and $CD$. Given lengths:

1. **Problem statement:** Given two parallel lines $m \parallel n$ cut by a transversal, find the values of $x$ and $y$ given the angles: - On line $m$: angle $= (8x + 13)^\circ$

1. **Problem Statement:** Find the measures of angles $m\angle 1$ through $m\angle 48$ given the polygon with vertices and internal polygons: a regular hexagon BCEFJ, a regular pen

1. **State the problem:** We want to find the total length $L$ of the string in a ball of string with radius $r = 2$ m. 2. **Formula and assumptions:** The ball is roughly a sphere