📐 geometry

1. **Stating the problem:** We have triangle ABC with sides $AB=4$ cm, $BC=6$ cm, and $AC=3.5$ cm. Points $E$ and $F$ are midpoints of sides $AB$ and $AC$ respectively.

1. **Plotting points A-H in 3D space:** - Points: A(0,0,2), B(0,2,0), C(2,0,0), D(2,3,0), E(3,2,4), F(-2,0,4), G(-1,-1,-1), H(2,1,-2).

1. **State the problem:** We need to find the scale factor of enlargement from the smaller shape to the larger shape in each case. 2. **Formula and rule:** The scale factor of enla

1. The problem states that the opposite angles of a parallelogram are given as $(2x - 3)^\circ$ and $(45 - x)^\circ$. We need to find the value of $x$. 2. Important property: Oppos

1. **State the problem:** We have three congruent rectangles arranged so that each is rotated 90° around a vertex of the previous rectangle. Given points A(2,5) and B(4,1.5), find

1. **Problem statement:** We have a quadrilateral OXPY with diagonals OX and PY intersecting at point T. Given:

1. **Stating the problem:** We are given a circle with center O and a central angle $\alpha = 288^\circ$. Points A, B, C, D, and H lie on the circumference, and angles $x$, $y$, an

1. **State the problem:** Calculate the area of the shaded region formed by a large quarter circle of radius 12, a smaller quarter circle of radius 5, and a right triangle with leg

1. The problem asks which of the given shapes (A, B, or C) has rotational symmetry. 2. Rotational symmetry means a shape looks the same after being rotated by some angle less than

1. **Problem Statement:** We have triangle ABC with angle $\angle A = 45^\circ$ and a segment marked as $2\alpha$ near base BC. Point W lies on BC. We want to solve for unknowns or

1. **Problem statement:** We have an equilateral triangle ABC, and side BC is extended to point E such that C is the midpoint of BE. We need to find the measures of angles $\angle

1. مسئله: پنجره‌ای داریم که شامل یک مستطیل به عرض $x$ و ارتفاع $y$ است و بالای آن یک مثلث متساوی الاضلاع با ضلع $x$ قرار دارد. محیط کل پنجره برابر 8 متر است. می‌خواهیم طول ضلع مثلث

1. مسئله را بیان می‌کنیم: مثلثی با رئوس A(-1,2)، B(3,0)، C(1,-2) داده شده است. الف) معادله میانه AM را بیابید.

1. **Problem statement:** Given a right isosceles triangle $\triangle ABC$ with the right angle at $A$, $M$ is the midpoint of $BC$, and $E$ lies on segment $MC$. Lines $BH$ and $C

1. **Problem Statement:** Determine if triangles ABC and DEF are similar given some side lengths and angles, and if possible, find the length of side EF. 2. **Similarity Criteria:*

1. **Problem:** Find the nets of the following solids: cube, cuboid, prism, cylinder, cone, and sphere. 2. **Understanding Nets:** A net is a two-dimensional shape that can be fold

1. The problem states that quadrilateral PQMN lies on one circle and quadrilateral SRMN lies on another circle. 2. This means PQMN and SRMN are cyclic quadrilaterals.

1. **Problem Statement:** We are given two circles intersecting at points $M$ and $N$. Points $P$, $Q$, $R$, and $S$ lie such that $PQRS$ is a straight line. Two triangles $PNR$ an

1. مسئله: نشان دهید مثلث‌های CDE و ABC متشابه هستند و سپس طول ضلع BC را با توجه به داده‌ها محاسبه کنید. 2. برای اثبات تشابه مثلث‌ها، از قانون تشابه مثلث‌ها استفاده می‌کنیم که می‌گو

1. مسئله: در مثلث قائم الزاویه با داده‌های $AC=6$ و $HC=4$، باید طول‌های $BC$ و $AB$ را پیدا کنیم. 2. ابتدا باید بدانیم نقطه $H$ روی کدام ضلع قرار دارد. فرض می‌کنیم $H$ روی ضلع $AC

1. **Problem Statement:** Given a set of vertices and distances between them, we want to analyze the relationships or possibly find missing distances or verify consistency. 2. **Kn