📐 geometry

1. **State the problem:** We need to find the value of angle $x$ in quadrilateral $ABCD$ composed of two triangles $ABC$ and $BCD$ with given sides and angles. 2. **Given data:**

1. **State the problem:** We are given a right-angled triangle AOC with AC = 24 cm, OC = 26 cm, and angle \(\angle AOC = 60^\circ\). Inside it is a children's swimming pool shaped

1. **Problem statement:** We have an equilateral triangle formed by towns L, M, and N. The bearing of M from L is 103°. We need to find the bearing of L from N. 2. **Understanding

1. **Problem statement:** Prove using vectors that the diagonals of a rhombus bisect each other. 2. **Setup:** Let the rhombus have vertices represented by vectors \( \vec{A}, \vec

1. Stating the problem. A wall 10 m long, 20 cm thick and 1.2 m high is constructed as a partition between two plots of land.

1. **State the problem:** We have a wall with dimensions length = 10 m, thickness = 20 cm = 0.2 m, and height = 1.2 m. We need to find the cost of cementing five sides of this wall

1. **State the problem:** We need to find the area of an L-shaped polygon composed of two rectangles and a triangular cutout. 2. **Identify the shapes and dimensions:**

1. The problem is to explain the mensuration formulas for a cylinder and a sphere. 2. For a cylinder, the important measurements are its radius $r$ and height $h$.

1. **Problem Statement:** Given a circle with diameter \(\overline{EF}\) and tangent \(\overline{EH}\) at point \(E\), and the measure of arc \(\widehat{EFG} = 204^\circ\), find:

1. **Problem statement:** Given quadrilateral ABCD with AB = DC and \(\angle ABC = \angle BCD\) (both acute), prove that AD is parallel to BC. 2. **Key information:**

1. **State the problem:** We have a right triangle with legs $CA = \sqrt{5} - 2$ and $CB = \sqrt{5} + 2$, and we need to find the hypotenuse $AB$. 2. **Formula used:** In a right t

1. **Problem 1: Given polygons ABCH and EDCH are congruent, with C on BD, m(\angle AHC) = 110^\circ, BC = 5 cm. Find the requested values.\n\n2. Since ABCH \cong EDCH, correspondin

1. **Problem Statement:** We have polygon ABCD reflected about the y-axis to form polygon EFGD. Given angles at vertices A (120°) and G (60°), and right angles at B, C, F, and G, w

1. **Problem Statement:** Given two congruent polygons ABCD and AFED, with C as the midpoint of segment MD, and MC = 3 cm, find the length of AE.

1. **Problem statement:** We have two polygons ABCD and EFGH. Polygon EFGH is the image of ABCD by a translation.

1. **Problem Statement:** We have two congruent polygons ABCD and FECD sharing side CD. Given perimeter of ABCD is 24 cm and CD = 5 cm, find perimeter of polygon ABEFD. 2. **Unders

1. The problem is to draw a 10 degree angle and label it. 2. An angle is formed by two rays with a common endpoint called the vertex.

1. **Problem:** Given a circle with radius $8$ m and central angle $45^\circ$, find the arc length. 2. **Formula:** The arc length $s$ of a circle is given by

1. مسئله: دو ضلع مثلث به طول‌های $2$ و $5$ داده شده‌اند و زاویه بین آن‌ها بین $0$ و $90$ درجه است. مساحت مثلث برابر $3$ است. باید کسینوس زاویه بین دو ضلع را پیدا کنیم. 2. فرمول مسا

1. **Problem Statement:** If the diagonals of a parallelogram are equal, prove that the parallelogram is a rectangle.

1. **Problem Statement:** Given trapezium ABCD with AB || CD, E and F are midpoints of non-parallel sides AD and BC respectively. We need to prove that EF is parallel to AB and EF