📐 geometry

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

1. **Problem Statement:** Given points $P(-4,-1)$, $Q(6,3)$, $R(6,b)$, and $S(-4,-3)$, we need to solve several parts related to gradients, parallelism, lengths, midpoints, and pro

1. **Problem Statement:** Prove that a cyclic parallelogram is a rectangle. 2. **Key Definitions:**

1. **Problem Statement:** We have a rectangle with vertical side length $2400$ mm and horizontal side length $2200$ mm. We want to find the length of the diagonal, represented by t

1. **State the problem:** We have a line segment \(\overline{AB}\) with endpoints \(A(1, 3)\) and \(B(5, 3)\).

1. **State the problem:** We have a triangle ABC with vertices A(-4, 3), B(2, 3), and C(-5, 1). We apply a dilation centered at the origin with a scale factor of 5. 2. **Formula fo

1. **Stating the problem:** We have a rectangle with height $2400$ mm and width $2200$ mm. We want to find the length of the diagonals and the point where they intersect. 2. **Form

1. **Problem Statement:** We have a triangle ABC with points A(1,1), B(2,3), C(4,1) that is dilated from the origin to triangle A'B'C' with points A'(3,3), B'(3,9), C'(12,3).

1. **Problem Statement:** We have triangle $\triangle LMN$ with vertices $L(2, 2)$, $M(6, 2)$, and $N(0, 6)$. We want to find the coordinates of $L'$, $M'$, and $N'$ after a dilati

1. **State the problem:** We have a rectangle ABCD with vertices A(2,1), B(2,4), C(6,4), and D(6,1). We need to find the coordinates of A', B', C', and D' after a dilation centered