📐 geometry

1. The problem asks to find the range of possible lengths for the third side $x$ of a triangle given two sides, using the triangle inequality rule: $$|a-b| < x < a+b$$

1. The problem states that the map scale is 1:4,000,000, meaning 1 cm on the map represents 4,000,000 cm in reality. 2. The distance between two cities on the map is 5 cm.

1. The problem asks if the given sets of side lengths can form a triangle. 2. Recall the triangle inequality theorem: For any triangle, the sum of the lengths of any two sides must

1. **State the problem:** We are given that \(\angle 1 \cong \angle 2\) and we want to prove that lines \(a\) and \(b\) are parallel. 2. **Given:** \(\angle 1 \cong \angle 2\).

1. **State the problem:** Given that \(\angle 1 \cong \angle 2\), prove that lines \(a\) and \(b\) are parallel. 2. **Given:** \(\angle 1 \cong \angle 2\).

1. **State the problem:** We have a circle centered at point $P$ with diameters $\overline{AD}$ and $\overline{BE}$. We need to find the measure of the minor arc $\stackrel{\large{

1. **State the problem:** We have a circle centered at point $P$ with points $A$, $B$, $C$, $D$, and $E$ on the circumference. 2. Given that $\overline{AD}$ and $\overline{BE}$ are

1. **State the problem:** We need to find the measure in degrees of the major arc $\stackrel{\large{\frown}}{BAC}$ on a circle centered at point $D$, with points $A$, $B$, and $C$

1. **State the problem:** We have a circle centered at point $D$ with points $A$, $B$, and $C$ on it in clockwise order. We know the angles at the center: $\angle ADB = 7N + 12$, $

1. **State the problem:** We need to find the arc measure of the minor arc $\overset{\large{\frown}}{BC}$ in degrees. 2. **Identify given information:**

1. **State the problem:** We need to calculate the distance between two points A(50,32W) and B(50,32W) on the Earth. The Earth radius is given as 6400 km, and use $\pi=3.14$. 2. **

1. Stating the problem: We are given two points on Earth, \( A(50^{\circ}N, 32^{\circ}W) \) and \( B(50^{\circ}S, 32^{\circ}W) \), and need to find the distance between them. The E

1. **State the problem:** We want to find the total surface area of the region common to the three cylinders given by: $$x^2 + y^2 = 3^2$$

1. **State the problem:** We have two similar quadrilateral shapes, Q and R.

1. **Understand the problem:** We need to find the bearing from point C to point D on a square grid map. The bearing is the angle measured clockwise from the north direction to the

1. **State the problem:** Complete the sentence to describe the locus of points inside rectangle PQRS that are equidistant from sides PS and QR.

1. **State the problem:** We need to find the perimeter of a right-angled triangle with height 9 cm, base divided into segments of 2 cm and 12 cm, and the right angle between the h

1. We are given a right-angled triangle with the vertical leg of length $5$ meters and the horizontal leg of length $18$ meters. 2. To find the area of a right-angled triangle, we

1. **Énoncé du problème :** Dans un repère orthonormé $\left(O, \overrightarrow{i}, \overrightarrow{j}\right)$, on considère les points $A(3,-2)$, $B(0,-1)$, $C(4,1)$ et $D(-2,3)$.

1. Problem 21 Part 1: Point C (-1, 2) divides segment AB in the ratio 3:4, with A(2, 5). Find coordinates of B. 2. Use the section formula: If point C divides AB in ratio m:n, then

1. Problem 13: Given triangles ABC and DEF with $$\frac{AB}{DE} = \frac{BC}{FD}$$, find the condition for similarity. Step 1: For two triangles to be similar by SAS (Side-Angle-Sid