📐 geometry

1. **State the problem:** We have a square piece of cardboard 2 ft wide. We cut out a square of side length $x$ from each corner and fold up the sides to form an open box. We want

1. **State the problem:** We have a right-angled triangle ABC with right angle at B. Point P lies on BC such that BP = 4 cm and PC = 6 cm, so BC = 10 cm. The hypotenuse CA = 8 cm.

1. **State the problem:** We have a square divided into 4 horizontal strips (A, B, C, D) of equal height. Each strip is divided into vertical sections: A has 2, B has 3, C has 4, a

1. **State the problem:** We need to determine if the two shaded triangles on the square grid have the same area. 2. **Recall the formula for the area of a triangle:** The area $A$

1. **State the problem:** Amber wants to tile a bathroom wall measuring 1.2 meters by 2.16 meters using square tiles of side lengths 10 cm, 12 cm, or 18 cm. We need to determine wh

1. **Problem Statement:** Calculate the area of two irregular polygons. 2. **Understanding the first polygon:** It is a yellow irregular polygon with sides labeled $a$, $b$, and $c

1. The problem asks to explain the Flächeninhalt (area) of Figur 1 again. 2. Flächeninhalt means the area of a figure, which is the amount of space inside its boundary.

1. Let's start by stating the problem: We want to understand how to calculate the area of a shape. 2. The formula for area depends on the shape. For example, the area of a rectangl

1. نبدأ بتحديد الزوايا المعطاة داخل الدائرة: 90، 30، 100، 90، 30، و5 درجات. 2. نعلم أن مجموع الزوايا حول نقطة داخل الدائرة يجب أن يكون 360 درجة.

1. The problem is to plot two points A(1,-2,2) and B(2,-2,-3) in a 3D coordinate system with X, Y, and Z axes. 2. Point A has coordinates $x=1$, $y=-2$, and $z=2$. This means it is

1. The problem states we have a point $A(1,-2,2)$ in 3D space. 2. Since no specific question is asked, let's analyze the point's coordinates.

1. Задача: Знайти об'єм паралелепіпеда, заданого точками $A(-2;5;3)$, $B(8;7;3)$, $C(-9;1;4)$, $A_1(-5;-1;5)$. 2. Об'єм паралелепіпеда можна знайти як абсолютне значення скалярного

1. The problem involves a square OABC with vertices O(0,0), A(3,0), B(3,3), and C(0,3). 2. We first consider the translation of the square by the vector $\begin{pmatrix}2 \\ 2\end{

1. The problem gives two teardrop-shaped closed curves with areas and perimeters: - Bottom shape: area $A_1 = 7$ cm$^2$, perimeter $P_1 = 9$ cm

1. **State the problem:** We have a solid cone with radius $r=5$ cm and perpendicular height $h=12$ cm.

1. **State the problem:** Find the distance between points $A(2,3)$ and $B(7,15)$.\n\n2. **Recall the distance formula:** The distance $d$ between two points $(x_1,y_1)$ and $(x_2,

1. Problem: If TP and TQ are tangents to a circle with center O such that $\angle POQ = 110^\circ$, find $\angle PTQ$. Step 1: Recall that tangents from an external point are equal

1. **State the problem:** We need to find the equation of the locus of a point $P(x,y)$ that is always 5 units away from the origin $(0,0)$. 2. **Understand the concept:** The locu

1. The problem involves finding the area of a trapezoid with given dimensions. 2. The trapezoid has a top base of length $7$, a height of $2$, and a bottom base with endpoints labe

1. **State the problem:** We have a square divided into four quadrilaterals by its two diagonals intersecting inside it. The areas of three quadrilaterals are given: 42 cm² (top-le

1. **Stating the problem:** We have two similar tins with masses 2000g and 500g respectively. The area of the base of the smaller tin is 100 square centimeters. We need to find the