📐 geometry

1. We are given a right trapezoid with one base of unknown length, a vertical height of 3 cm, a diagonal side of 12 cm, and a small inscribed right triangle with height 2 cm. 2. Le

1. The problem states: Find the perimeter of a square with side length 12 cm. 2. The formula for the perimeter $P$ of a square with side length $s$ is:

1. **State the problem:** We are given a right triangle ABC with right angle at B. We know \( \cos C = \frac{4}{5} \), vertical side CB = 16, and a segment ED inside the triangle p

1. **State the problem:** We have a right-angled triangle with the hypotenuse length $\sqrt{48}$ cm and one side adjacent to angle $x$ of length $\sqrt{12}$ cm. We need to show tha

1. **State the problem:** We are given that ABC is a straight line parallel to DF, with BD = DE.

1. **Problem (a):** Calculate the side $x$ opposite the $30^\circ$ angle in a right triangle with hypotenuse $12$ cm.

1. **Problem statement:** Given that AD, BE, and CF are parallel lines and IJ is parallel to GH, we are to find the value of \(\angle JIH\). The angle \(\angle GHI = 80^\circ\) and

1. **Problem statement:** Count how many pairs of lines (edges) in the cuboid can intersect when extended in 3D.

1. Stating the problem: We have triangle ABC with points D and E on segment AD, where segment EB is parallel to segment DC. 2. Given: $AB = 12$, $AC = 16$, and $ED = 5$.

1. The problem involves exploring an infinite iterative pattern alternating between equilateral triangles and their circumscribed circles. 2. Given the base equilateral triangle wi

1. The problem is to verify the correctness of given geometric patterns and area formulas for alternating equilateral triangles and circles that are inscribed and circumscribed wit

1. State the problem: We need to find the value of $x$ such that the volume of a rectangular solid with dimensions 2, 8, and $x$ equals the volume of a cube with side length 4. 2.

1. **State the problem:** We are given that \(\angle ABC = \angle ACB\) (the angles are equal). The lengths are given as \(AE = 3x + 4\), \(CE = y - 3\), \(BE = 2x + 1\), and \(DE

1. The problem involves studying the iterative pattern formed by an equilateral triangle and its circumscribed circle, each inscribed within the other repeatedly. 2. First, define

1. **Problem statement:** Find the area of a single triangular petal in the second iteration of the Diamond Symbol of the Daisy Petals figure. 2. **Given:** The base triangle area

1. **Restate the problem:** We are investigating the iterative pattern formed by inscribing triangles and circles inside each other repeatedly. 2. **Preliminary understanding:** Ea

1. The user requests Chapter 2 titled "Definition of Terms, Preliminary Concepts and Conjuncture." 2. This chapter typically defines key terms used in the study, explains foundatio

1. The problem is to find the total area $AT$ of an irregular polygon composed of two rectangles and a right triangle. 2. First, identify the areas given: $AD$ is the sum of the tw

1. **Problem 49:** Given a right triangle with perpendicular side $9$ cm, hypotenuse $12$ cm, and segments $x, y, z$ on the adjacent side and altitude to hypotenuse, find $x + y +

1. Let's start by stating the problem: We want to understand how to use similarity in triangles to solve problems. 2. Similar triangles have the same angles and their corresponding

1. **State the problem:** Calculate the area of the shaded part inside triangle ABC where M is the intersection of the medians. 2. **Recall the property of medians:** The medians o