📐 geometry

1. **State the problem:** We have a circle with radius $14$ cm and two chords $AB$ and $CD$ with lengths $12$ cm and $10$ cm respectively. We need to find the distance of each chor

1. **Problem statement:** We have a circle with center $O$ and points $A$, $B$, $C$, and $D$ on its circumference. A tangent line $FDE$ touches the circle at point $D$. We need to

1. **State the problem:** We need to find the size of angle $\angle DAC$ in a circle with diameter $AC$, where $\angle CAB = 25^\circ$ and $\angle DEC = 100^\circ$. 2. **Recall key

1. **State the problem:** We have points A, B, C, D on a circle, with C, D, E collinear. Given that $BA = BD$, $CB = CD$, and angle $ABD = 40^\circ$, we need to find the size of an

1. **Problem statement:** We have triangle RUZ with sides $RU=47$ cm, $RZ=102$ cm, and angle $Z=23^\circ$. We need to find the measure of the obtuse angle $RUZ$ (angle at $U$). 2.

1. **Problem statement:** We have two parallel lines $a$ and $b$ with a 45° right triangle placed between them. Given that angle $\angle 1 = 15^\circ$, we need to find the size of

1. **Problem Statement:** Use the Pythagorean theorem to find the length of the hypotenuse $c$ in a right triangle with legs $a$ and $b$. 2. **Formula:** The Pythagorean theorem st

1. **Stating the problem:** We have a geometric figure with angles 45°, 80°, and unknown angles $a^\circ$, $b^\circ$, $c^\circ$, and $d^\circ$. We need to find the values of $a$, $

1. **State the problem:** Calculate the perimeter and area of the given irregular "T"-shaped polygon with dimensions 10 cm and 4 cm. 2. **Perimeter calculation:** The perimeter is

1. **State the problem:** We have a cuboid with length $L$ cm, width $W$ cm, and height $H$ cm. The net of the cuboid shows dimensions: total horizontal length $37.8$ cm and vertic

1. সমস্যাটি হলো: একটি বৃত্তে ABC ত্রিভুজের কোণ $\angle ABC = 30^\circ$ এবং বৃত্তের ব্যাসার্ধ $5$ মিটার। আমাদের অর্ধবৃত্তের ব্যাসার্ধ নির্ণয় করতে হবে। 2. সূত্র: বৃত্তের ব্যাসার্ধ $

1. **সমস্যা:** একটি সমবাহু ত্রিভুজের প্রতি এক বাহুর দৈর্ঘ 4 মিটার বাড়ালে ক্ষেত্রফল 20\sqrt{3} বর্গমিটার হয়। ত্রিভুজটির মূল বাহুর দৈর্ঘ্য নির্ণয় কর। 2. **সূত্র:** সমবাহু ত্রিভুজে

1. 問題陳述：計算一個圓形噴水池的面積，已知圓周長為 37.68 m，取 π = 3.14。 2. 公式與規則：

1. **Nêu bài toán:** Cho tam giác nhọn ABC với các đường cao AD, BE, CF. Gọi K, L lần lượt là tâm đường tròn nội tiếp các tam giác CDE, BDF. Gọi P, Q lần lượt là tâm đường tròn ngo

1. **Problem Statement:** Find the values of angles $a$, $b$, $c$, and $d$ in the circle with center $O$ where $PQ$ is tangent at $A$. Given angles: $\angle C = 110^\circ$, $\angle

1. **Problem 5:** Given a circle with center O, tangent PQ at A, and angles marked as $110^\circ$ at C, $60^\circ$ at D, and unknown angles $a$, $b$, $c$, and $d$. Find $a$, $b$, $

1. **Problem statement:** We have an isosceles triangle with two equal sides of length 8 cm and an included angle of 55° between them. We need to find the angle $x$ at the top vert

1. **State the problem:** We have an isosceles triangle PQR symmetrical about axis m, with angle Q measuring 35°. We need to find the measure of angle XPR.

1. **State the problem:** We need to find the perimeter of a right triangle with a height of 9 cm perpendicular to the base, which is divided into two segments of 4 cm and 12 cm. 2

1. **State the problem:** We are given a right-angled triangle with two shorter sides (legs) measuring 14 cm and 26 cm. We need to find the length of the hypotenuse. 2. **Formula u

1. **State the problem:** We have three similar solid cylinders L, M, and P made from the same material. Given masses and surface areas, we need to find the ratio of their heights: