📐 geometry

1. **Problem statement:** Given that $ABC$ and $BED$ are straight lines, find the values of the unknown angles $a$ and $b$ in the diagram where $\angle BED = b^\circ$, $\angle AEB

1. **Problem statement:** We have a trapezoid with the following dimensions: top base $= 12.0$ cm, bottom base $= 6.0$ cm, left side $= 4.3$ cm, right side $= 6.0$ cm, and height $

1. Let's start by stating the problem: We want to understand how to convert between Cartesian coordinates $(x,y)$ and polar coordinates $(r,\theta)$. 2. The formulas for conversion

1. The problem is to convert a point or equation from Cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$. 2. The formulas for conversion are:

1. The problem: Explain the Second Pappus’ Theorem, which relates the surface area of a solid of revolution to the centroid of the generating curve. 2. Statement: The Second Pappus

1. The problem: Explain the First Pappus’ Theorem, which relates the surface area of a solid of revolution to the path traveled by the centroid of the generating curve. 2. Statemen

1. The problem: Understand what quadrants are in the coordinate plane. 2. The coordinate plane is divided into four sections called quadrants by the x-axis (horizontal) and y-axis

1. Let's understand what Cartesian coordinates are. They represent a point in a plane using two numbers: the $x$-coordinate (horizontal position) and the $y$-coordinate (vertical p

1. **Stating the problem:** Find the angles DON, DNQ, DBA, ONA, and ODN in the given circle with center O, tangent PQ at N, and points A, B, C, D on the circumference.

1. **Problem statement:** In triangle $\triangle ABC$, $D$ is the midpoint of side $BC$. Point $E$ lies on side $AC$ such that $CE = \frac{AC}{3}$. Lines $BE$ and $AD$ intersect at

1. समस्या: त्रिभुज ABC का केन्द्रक (centroid) G(4, 5) है, और शीर्ष A के निर्देशांक (2, 3) हैं। बिंदु D, भुजा BC का मध्य-बिंदु है। हमें D के निर्देशांक ज्ञात करने हैं। 2. सूत्र: केन

1. **Problem Statement:** We have 6 pink tiles and 6 green tiles, each triangular with given side lengths.

1. **Problem Statement:** We have two adjacent triangular fields. The first triangle has sides 100 m, 70 m, and 50 m. The second triangle has sides 30 m, 80 m, and 70 m. We need to

1. **Problem Statement:** Find the area of quadrilateral PQRS by dividing it into two triangles PQS and QRS and using Heron's formula. 2. **Heron's Formula:** For a triangle with s

1. **Problem Statement:** Given a quadrilateral ABCD where $AC=BC=DC$, $\angle ADB=30^\circ$, and $\angle CAD=10^\circ$, find $\angle ACB$.

1. **Problem statement:** We need to find the length of chord AB in a circle with diameter 14, where the angle between the radius line and the chord is 30°. 2. **Given:** Diameter

1. **State the problem:** We need to find the average area of all circles with radii between 3 and 6 meters. 2. **Formula for the area of a circle:** The area $A$ of a circle with

1. **Problem Statement:** We have a circle with an inscribed triangle. One side of the triangle is 14 units, and another side is 39 units. We need to find the length of the third s

1. **State the problem:** We need to find the diameter of a circle given its circumference is 400 meters. 2. **Formula used:** The circumference $C$ of a circle is related to its d

1. **Problem Statement:** We have two chords intersecting inside a circle. One chord is divided into two segments of lengths 14 and 64. We want to find the length of the other chor

1. مسئله: در مثلث ABC، زاویه BÂD برابر با زاویه DÂC است. داده شده AB = 10، AE = 6 و DE = 4. باید اندازه BD را پیدا کنیم. 2. چون BÂD = DÂC، مثلث ABD و مثلث ADC دو زاویه برابر دارند