📐 geometry

1. The problem involves using the Pythagorean theorem, which relates the lengths of the sides of a right triangle. 2. The Pythagorean theorem formula is $$a^2 + b^2 = c^2$$ where $

1. **State the problem:** We need to find the length $x$ in a trapezoid where the top base is 9, the bottom base is 31, and the diagonal is 19. A perpendicular line $x$ is drawn fr

1. **Stating the problem:** We have triangle vertices at points $A(2,2)$, $B(8,2)$, and $C(5,5)$. We want to find the new coordinates after rotating the triangle $90^\circ$ anticlo

1. **State the problem:** We have a convex pentagon with interior angles measuring $2v$, $2v + 43^\circ$, $2v + 12^\circ$, $96^\circ$, and $v + 25^\circ$. We need to find the value

1. **State the problem:** We need to find the number of sides $n$ of a regular polygon given that each interior angle measures 140°. 2. **Formula for interior angle of a regular po

1. **State the problem:** We need to find the volume of one hemisphere of the Earth, which is roughly spherical with a diameter of 12,800 km. 2. **Formula used:** The volume of a s

1. **State the problem:** Find the distance between the points $(-4,8)$ and $(4,4)$. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given b

1. **Problem statement:** Find the volume of water in the cylinder with a metal ball bearing fully submerged inside it. 2. **Given:**

1. **Problem Statement:** Identify similar triangles in the given figures and solve for missing sides using right triangle similarity and geometric mean theorems. 2. **Similar Tria

1. **Énoncé du problème :** On considère un triangle ABC avec (EF) parallèle à (BC), et les longueurs suivantes : $AC=4$, $BC=5$, $EA=12$, $AB=3$.

1. **Stating the problem:** We want to find the coordinates of point P that divides the line segment AB in the ratio $m:n$.

1. **Problem statement:** Given circle Q with \(\overline{MN} \cong \overline{OP}\) and radius 12 units, find the measure of \(\angle 4\). 2. **Relevant formula and rules:** In a c

1. **Stating the problem:** We need to find the measure of angle $\angle xyb$ given that $\angle xyb$ is formed by a line segment from point $y$ to $x$ making a $15^\circ$ angle wi

1. **Stating the problem:** We are given a horizontal line with points A, C, and B, and a line from point C forming a 60° angle with the horizontal. We need to find the measure of

1. **State the problem:** We are given an arc length $s = 30$ cm and a central angle $\theta = 120^\circ$ in a circle. We need to find the radius $r$ of the circle. 2. **Formula us

1. **Problem statement:** We have a circle with center $O$ and a circumscribed angle $A$ around it. We want to find the length of segment $\overline{AB}$. Given are an angle of $73

1. **Problem statement:** We have a circle with center $O$ and a circumscribed angle $A$ around it. We want to find the length of segment $\overline{AB}$. Given are an angle of $73

1. The problem asks for the coordinates of the two labeled points on the graph. 2. From the description:

1. **Problem:** What is an angle whose vertex is on a circle and whose sides contain chords of the circle? 2. **Definition:** An inscribed angle is an angle whose vertex lies on th

1. **Problem statement:** We have an equilateral triangle $\triangle ABC$ inscribed in a circle $\omega_1$ with radius 4. Another circle $\omega_2$ with radius 2 is tangent interna

1. **State the problem:** Find the midpoint of the line segment joining the points (8, -6) and (4, 5). 2. **Formula:** The midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and