📐 geometry

1. **State the problem:** We have a semicircle with diameter 10 and a rectangle inscribed inside it. One corner of the rectangle is at the midpoint of the diameter, and the opposit

1. **Identify circle parts in \( \emptyset O \):**\n\n- Radii: segments from center \( O \) to circle points, e.g., \( OJ \) and \( OS \).\n- Diameters: chords through \( O \) pass

1. First, let's restate the problem in English:\nWe need to find the minimum length of iron rod needed to make 100 window grilles shaped as circular sectors (juring) with radius $1

1. The problem describes a coordinate system with an angle $\theta$ formed at the origin between the $x$-axis and a ray extending upward right. 2. Points $N$, $M$, and $M'$ lie on

1. The problem describes two coordinate systems sharing the same origin $O$, with axes $X, Y$ and rotated axes $X', Y'$. 2. The angle between the original $X$ axis and the rotated

1. The problem involves finding the equation of the line of reflection (mirror line) for triangle POR reflected to triangle P'Q'R'. 2. Given points are P(6,2), R(9,2) in the origin

1. **Stating the problem:** We are given a triangle PQR with angles \(x^\circ\), \((x + 30)^\circ\), and \(3x^\circ\). We need to find the value of \(x\).\n\n2. **Using the triangl

1. **Stating the problem:** We have a right triangle where the vertical side is 6 units, and the horizontal side is given as 10 feet 8 inches. The angle adjacent to the horizontal

1. **State the problem:** Determine which sequences of transformations prove congruency by mapping polygon I onto polygon II. 2. **Analyze Shape Locations:** Polygon I is in quadra

1. The problem involves determining the values of angles \(\angle 1\) through \(\angle 27\) in a circle with many chords and points. 2. Given the complexity and the provided formul

1. **Problem statement:** Points A and B lie on the circumference of a circle with center O and radius $r$. The angle $\angle AOB = \theta$ radians and $C$ is the midpoint of $OA$.

1. The angle formed by a tangent and a secant outside a circle is half the difference of the intercepted arcs. Here, arcs are 200° and 220°. $$x=\frac{|220-200|}{2}=\frac{20}{2}=10

1. **Problem Statement:** We analyze Diagram B and solve the questions: identify triangles, check for congruency, complete the transformation table, and discuss similarity and cong

1. সমস্যা বর্ণনা: একটি সামান্তরিকের দুইটি বাহুর দৈর্ঘ্য যথাক্রমে 30 সে.মি. এবং 26 সে.মি. । তাদের মধ্যকার ক্ষুদ্রতম কোণ 28 ডিগ্রি। 2. আমরা জানতে চাই সামান্তরিকটির ক্ষেত্রফল।

1. সমস্যাটি হলো, একটি সামান্তরিকের বাহুর দৈর্ঘ্য ৩০ সেমি এবং ২৬ সেমি। ক্ষুদ্রতম কণ্টি ২৮ সেমি হলে অপর কণ্টির দৈর্ঘ্য নির্ণয় করতে হবে। 2. সামান্তরিকের বাহুর দৈর্ঘ্য: $a = 30$ সেমি,

1. **Problem statement:** We have a tent shaped as a triangular prism with base triangle sides 2.5 m, 2.5 m, and 1.5 m height, length of prism 6 m, and base of prism 4 m. We need t

1. Problem: Calculate the total calico material needed (surface area) and volume of a tent shaped like a triangular prism. 2. Given dimensions:

1. **Find the size of angle $\Theta$ to the nearest degree:** The problem gives two sides adjacent to $\Theta$ in a right triangle: opposite side length 21 cm and adjacent side len

1. **State the problem:** We have a tent shaped as a triangular prism with triangular faces of sides 2.5 m, 2.5 m, and 4 m. The height of the triangle is 1.5 m, and the length (dep

1. The problem asks us to find the total amount of calico material needed to make a tent shaped like a triangular prism, including its ground sheet, with the total surface area giv

1. **Stating the problem:** We have a triangle with side AB length 90 feet, angle A = 60°, angle B = 30°, and the right angle opposite side AB. We want to find the distance $i$ fee