📐 geometry

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

1. **Problem statement:** Two towers of heights 30 m and 40 m are 50 m apart. A well is located between them on the line joining their bases. Two birds fly simultaneously from the

1. **Problem statement:** Given a circle with center O, chord PQ of length 8 cm, N is the midpoint of PQ, ON = 3 cm, and \(\angle ONR = 20^\circ\). We need to find the size of \(\a

1. The problem asks for the total area of a rectangular rug with width 8 meters and height 2 7/12 meters. 2. First, convert the mixed number height to an improper fraction:

1. The problem asks to translate shape A, a trapezoid located approximately between $x = -7$ to $x = -3$ and $y = 3$ to $y = 6$, by the vector $\begin{pmatrix}8 \\ -2\end{pmatrix}$

1. **State the problem:** We are given quadrilateral RSTU with angles and side lengths, and we need to find the measures of angles $\angle RST$ and $\angle RTU$ using the given inf

1. **Stating the problem:** We have a quadrilateral R S T U with diagonal S U. Given angles are \(\angle R = 30^\circ\) and \(\angle R S U = 25^\circ\). The segment \(R U = 4b - 1\

1. **State the problem:** We have quadrilateral RSTU with given angles and side lengths. We need to find the measures of angles $\angle RST$ and $\angle RTU$.

1. The problem asks to identify all triangles similar to triangle T. 2. Similar triangles have the same shape but may differ in size; they have equal corresponding angles and propo