📐 geometry

1. **Nêu bài toán:** Cho tam giác ABC vuông tại A. Gọi M là trung điểm của cạnh BC. Trên tia đối của tia AM lấy điểm E sao cho ME = MA. Chứng minh: a) Tam giác AMB = tam giác EMC

1. **State the problem:** Find the farthest distance from the point $(12, 2)$ to the circle given by the equation $$x^2 + y^2 + 6x - 16y + 24 = 0.$$\n\n2. **Rewrite the circle equa

1. **Problem 1: Identify the hypotenuse in a spherical triangle with angles** $A=90^\circ$, $B=100^\circ$, $C=92^\circ$. 2. In spherical triangles, the side opposite the largest an

1. **State the problem:** Determine if a spherical triangle can be constructed with sides $a=170^\circ$, $b=100^\circ$, and $c=80^\circ$. 2. **Recall the spherical triangle inequal

1. **Problem Statement:** Given a spherical triangle with angles $A=130^\circ$, $B=110^\circ$, and $C=140^\circ$ on a sphere of radius $R=15$ m, determine the spherical excess.

1. **Stating the problem:** We have a truncated regular quadrilateral pyramid (a frustum) with height $h=6$ cm.

1. **Problem statement:** Given a circle $(O)$ with radius $R=6$ cm and a point $E$ outside the circle such that $OE=9$ cm. Two tangents $EA$ and $EB$ touch the circle at points $A

1. **Problem statement:** Given a circle $(O)$ with radius $R=6$ cm and a point $E$ outside the circle such that $OE=9$ cm. Two tangents $EA$ and $EB$ touch the circle at points $A

1. সমস্যাটি হলো: সমকোণী ত্রিভুজ ABC তে, যেখানে \(\angle B = 90^\circ\) এবং বাহু AC = 10 সেমি। ত্রিভুজটির বাহুগুলির দৈর্ঘ্যের বর্গের সমষ্টি কত? 2. সূত্র: সমকোণী ত্রিভুজের ক্ষেত্রে,

1. **Nêu bài toán:** Từ điểm $M$ nằm ngoài đường tròn $(O,R)$ sao cho $OM=2R$. Vẽ hai tiếp tuyến $MA$, $MB$ với đường tròn $(O)$, $A$, $B$ là hai tiếp điểm. Gọi $H$ là giao điểm củ

1. مسئله: مجموع زوایای داخلی یک چندضلعی منتظم با 15 ضلع را پیدا کنید. 2. فرمول مجموع زوایای داخلی هر چندضلعی با $n$ ضلع به صورت $$180(n-2)$$ درجه است.

1. The problem is to find the value of angle $x$ in a triangle where the other two angles are $115^\circ$ and $35^\circ$. 2. Recall the Triangle Angle Sum Theorem: The sum of the i

1. The problem is to create assertion-reason type questions for the chapter "Construction of Triangles" for class VIII. 2. Assertion-Reason questions consist of two statements: an

1. **State the problem:** Find the equations of the altitudes of the triangle with vertices A(0,4), B(4,6), and C(-2,-2). 2. **Recall the definition:** An altitude of a triangle is

1. **Problem statement:** In triangle ABC, given $\angle A = 40^\circ$, $\angle B = 80^\circ$, and side $b = 14.4$ feet, find $\angle C$ and side $c$. 2. **Step 1: Find $\angle C$*

1. مسئله: ما می‌دانیم که EB و BC و ABCD مربع هستند و مساحت EB برابر مساحت ABCD است. باید مساحت مثلث ABE را محاسبه کنیم.

1. The problem involves understanding the arrangement of eight circular logos in a grid of two rows and four columns, each representing a FIFA tournament. 2. The logos are arranged

1. **مشكلة:** لدينا دائرة نصف قطرها $3$ وحدات، ونريد حساب مساحة الجزء المظلل الذي يمثل قطاع دائري بزاوية $60^\circ$. 2. **الصيغة المستخدمة:** مساحة القطاع الدائري تُحسب بالعلاقة:

1. **Problem statement:** Given a rhombus ABCD with |AC| = |DC|, angles m(ABC) = 70°, m(ACB) = 40°, and m(BDC) = 75°, find the measure of angle m(BCD) = \(\alpha\). 2. **Recall pro

1. \textbf{Đề bài:} Cho tam giác ABC vuông tại A với AB<AC, AM là trung tuyến. Kẻ ME vuông góc với AB tại E, MF vuông góc với AC tại F. AM cắt EF tại G, CG cắt AB tại R. Chứng minh

1. **Énoncé du problème :** Nous avons un cercle de centre $O$ avec des points $A$, $B$, et $M$ sur le cercle.