📐 geometry

1. **State the problem:** We have two circles tangent to a horizontal line $\ell$ at points $S$ and $T$, with centers $P$ and $Q$ respectively. The circles touch each other at poin

1. **Problem statement:** We have a right-angled triangle BAC with right angle at A. Given lengths are $AB=12$ and $CD=1$. A semicircle with diameter $AD$ is tangent to side $BC$.

1. **State the problem:** We have a right trapezoid with the top base length 3, the right vertical leg divided into segments 2 and 6, and the bottom base length labeled as $x$. We

1. **State the problem:** We have an isosceles trapezium (trapezoid) with parallel sides of lengths 8 units and 18 units, and a circle inscribed inside it. We need to find the radi

1. **State the problem:** We have triangle \(\triangle A'B'C'\) with points \(A'(-5,-6)\), \(B'(-6,-2)\), and \(C'(-3,-3)\). We apply a glide reflection \(T_{(3,0)} \circ R_m\), wh

1. **Problem statement:** We have an isosceles trapezoid with parallel sides (bases) of lengths 8 units and 18 units, and a circle inscribed inside it. We need to find the radius o

1. **State the problem:** A square is cut into two identical rectangles. Each rectangle has a perimeter of 24 cm. We need to find the area of the original square. 2. **Define varia

1. **State the problem:** We have a trapezoid with a diagonal forming two right triangles. The top right triangle has legs 3 units and 2 units. The right side of the trapezoid is 6

1. The problem states that each of two rectangles has a perimeter of 24 cm. 2. Recall the formula for the perimeter of a rectangle: $$P = 2(l + w)$$ where $l$ is length and $w$ is

1. The problem asks for the volume of a sphere with radius $r = 10.5$ cm. 2. The formula for the volume of a sphere is $$V = \frac{4}{3} \pi r^3$$.

1. **State the problem:** We have a cube-shaped box with volume 8000 cm³. We need to find the length of one edge and the surface area of the cube using prime factorization. 2. **Re

1. **State the problem:** Construct quadrilateral PQRS with sides $PQ=4.5$ cm, $QR=6$ cm, $RS=9$ cm, $PS=6$ cm, and diagonal $QS=9$ cm. Find the measure of angle $\angle QRS$.\n\n2

1. **Énoncé du problème** : Placer le point $M$ sur le segment $[AB]$ tel que $AM = \frac{7}{5} AB$. 2. **Analyse** : Le segment $[AB]$ a une longueur $AB$. Le point $M$ doit être

1. Énonçons le problème : On doit placer un point $M$ sur le segment $[AB]$ tel que la longueur $AM$ soit égale à $\frac{7}{5}$ de la longueur $AB$. 2. Comprenons la condition : $A

1. **Problem 1:** A line from the center of a circle cuts a chord into two parts of lengths 3 cm and 2 cm. The perpendicular distance from the center to the chord is 1 cm. Find the

1. **State the problem:** We need to find the area and perimeter of two irregular polygonal shapes labeled 1 and 2 with given side lengths. 2. **Shape 1 area:**

1. **State the problem:** We have two irregular polygonal shapes with given side lengths. Shape 1 sides: 3 cm, 2 cm, 8 cm, 5 cm, 3 cm, 2 cm.

1. The problem asks for the distance of the point $(5,4)$ from the origin $(0,0)$. 2. The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ in the plane is given by the d

1. The problem states that triangle ABD is not equilateral. 2. An equilateral triangle has all three sides equal in length.

1. **Stating the problem:** We have a triangle ABC with an inscribed circle and an internal equilateral triangle with side length 2 cm. The triangle is extended to form a larger tr

1. **State the problem:** We are given two polygons on a coordinate plane with vertices A(0,4), B(2,3), C(4,1), D(4,-3), E(0,-4), F(-2,-3), G(-4,1), and H(-2,3). We need to find th