📐 geometry

1. **State the problem:** We are given a cube with volume $v = 729$ m$^3$ and side length $x = 9$ m. We need to calculate the surface area of the cube's net in cm$^2$ and express t

1. **Problem Statement:** Find the equation of a circle centered in the first quadrant that is tangent to the lines $x=8$, $y=3$, and $x=14$. 2. **Understanding the problem:** The

1. **State the problem:** Find the center and radius of a circle given the ends of a diameter at points $(18, -13)$ and $(4, -3)$. 2. **Formula used:**

1. **Problem statement:** Find the equation of a circle given its center and radius. 2. **Formula:** The standard form of a circle's equation with center at $(h,k)$ and radius $r$

1. **Problem statement:** We have a right triangle with legs $YZ = 20$ cm and $ZX = 21$ cm, and we want to find the hypotenuse $YX$. 2. **Formula used:** In a right triangle, the P

1. **Problem Statement:** We have a trapezoid with the top base $11$ cm, the bottom base $26$ cm, and one non-parallel side (left side) $17$ cm. We need to find the height $h$ of t

1. **State the problem:** We need to find the volume of air in a balloon with a diameter of 24 cm. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. **State the problem:** Prove that a parallelogram with congruent diagonals has a right angle. 2. **Given:**

1. **Problem:** Determine if the two triangles with one right angle and two equal sides are congruent. 2. **Rule:** Triangles with a right angle and two equal sides (hypotenuse and

1. **State the problem:** We need to find the volume of a composite rectangular prism composed of two sections: a bottom base and an upper section. 2. **Identify the dimensions:**

1. সমস্যা: △ABC এর অন্তঃস্থ একটি বিন্দু O দেওয়া আছে। প্রমাণ করতে হবে যে $$\angle BOC > \angle BAC$$। 2. সূত্র ও নিয়ম: একটি ত্রিভুজের অন্তঃস্থ বিন্দু হলো তিনটি কোণের সমদ্বিখণ্ডকগু

1. **Problem:** Given triangle ABC with sides AB = 9 cm, BC = 5 cm, AC = 8 cm, and segment BD = 6 cm, and angle at point E is 45°.

1. **Problem Statement:** Find the distance between each pair of points given. 2. **Formula:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the dis

1. **State the problem:** We have a triangle PQR with point T on side PR such that QT is perpendicular to PR. Given PR = 10 m, QT = 7 m, and angle PQT = 40°, we need to find the le

1. **Problem 9: Determine the area of the triangle given sides and angles.** 2. We use the Law of Sines formula to find missing side $x$:

1. **Problem 9:** Find the area of a right triangle with an angle of 64° and the side opposite this angle measuring 9 cm. 2. **Formula:** The area of a triangle can be found using

1. **State the problem:** We need to find the size of angle $k$ given a diagram with intersecting lines and angles $86^\circ$, $78^\circ$, and $102^\circ$. 2. **Recall the rule:**

1. **Problem statement:** We have puzzle pieces with central angles 70°, 32°, 136°, 58°, and one missing piece. These pieces should form a full circle (360°) and a semicircle (180°

1. **State the problem:** We need to construct a line segment AB of length 8 cm and its perpendicular bisector using a ruler and compass. Then, mark point C on the perpendicular bi

1. **State the problem:** We need to find the area of a quadrilateral that is divided into two right triangles by a diagonal. 2. **Identify the triangles and their sides:**

1. **Stating the problem:** We are given a right triangle EFB with perpendicular segments EF and FB, and EB perpendicular to HF. The lengths are $EF=\sqrt{20}$ units and $FH=2$ uni