📐 geometry

1. Problem statement: We need to prove the following relation involving segments of a cyclic quadrilateral ABCD with additional constructions: $$\frac{BE \cdot AH}{GH} = \frac{ED \

1. **Problem statement:** We have a right triangle with a hypotenuse of 105 ft making a 30° angle with the horizontal. The vertical side (height of the tower) is 85 ft, and a man s

1. **Nêu bài toán:** Cho tam giác ABC cân, các đường trung tuyến BM, CN cắt nhau tại G. Trên BG lấy điểm E là trung điểm của BG, trên CG lấy điểm F là trung điểm của CG. Yêu cầu:

1. **Problem 14:** In parallelogram ABCD, point P lies on CD such that AD = DP = PC. (i) To prove AP bisects \(\angle A\):

1. Problem: In the figure with circle and tangents, AC is tangent to the circle, BE is parallel to CD, and given angles are (a) $\angle ABE = 42^\circ$, $\angle BDC = 59^\circ$. Fi

1. **Problem 1:** Find the volume and total surface area of a right prism with bases 12 cm apart and equilateral triangular base with side 6 cm. 2. The height (distance between bas

1. Problem (i): Find the measure of \(\angle BOD\) in figure (ii).\nGiven vertical angles at intersection O, \(\angle BOD\) is opposite to the given \(80^\circ\) angle \(\angle AOC

1. Problem (18): The perimeter of a triangular field is 144 m and its sides are in the ratio 3 : 4 : 5. Find the area of the field. Step 1: Let the sides be $3x$, $4x$, $5x$. Since

1. The problem is to understand the basic properties and equations related to circle geometry. 2. A circle is defined as the set of points equidistant from a fixed point called the

1. **State the problem:** Determine the type of triangle ABC formed by points A(2,1), B(5,7), and C(8,4) by calculating the lengths of its sides. 2. **Find the length of side AB:**

1. **Problem statement:** We have a parallelogram ABCD with points O(1,0), A(2,5), B(8,8), C(11,5) and F inside it such that segment AF is perpendicular to OC. We want to verify th

1. The problem asks to translate a given triangle on the coordinate plane 4 units to the right and 1 unit down. 2. Translation means shifting every point of the figure by the same

1. The given equation is the Pythagorean theorem: $a^2 + b^2 = c^2$. 2. This theorem applies to right triangles, stating that the square of the hypotenuse ($c$) equals the sum of t

1. **State the problem:** We have a cylinder with radius $r = 3$ cm and volume $V = 250$ cm$^3$. We need to find the height $h$ of this cylinder. 2. **Recall the formula for the vo

1. **Calculate the area of the sector of a circle with radius 65 mm and sector angle 42°.** - The formula for the area of a sector is $$Area = \frac{\theta}{360} \times \pi r^2$$ w

1. The problem describes a semicircle with a radius of 7.2 meters and a horizontal diameter of 7.2 meters. This indicates possibly a misunderstanding since the diameter should be t

1. Problem statement: We are given an L-shaped polygon with a top horizontal edge of $12$ mm, an interior vertical located $4.3$ mm from the left that drops $2$ mm from the top to

1. **State the problem:** We are given an irregular polygon with labeled sides and need to find its area in mm². 2. **Analyze the shape:** The figure has two vertical segments labe

1. **Problem 10:** Calculate the perimeter of sector OAB with radius $r=7$ cm and area $40$ cm$^2$. 2. The area $A$ of a sector is given by $$A = \frac{1}{2} r^2 \theta$$ where $\t

1. **Problem 6**: A cone has a horizontal base diameter of 24 cm and an axis length of 36 cm with a volume of 3000 cm³. Find the angle of inclination of the axis with the horizonta

1. **State the problem:** We need to determine if a 50 cm ribbon is long enough to go all the way around the cake. 2. **Understanding the shape:** The top of the cake is a circle w