📐 geometry

1. **Stating the problem:** We are given a cuboid with a base rectangle having sides 4 cm and 19 cm, and a diagonal inside the base rectangle measuring 21 cm. We need to verify if

1. The problem is to understand the shape of the moon during its first or last quarter phase, which resembles a crescent shape. 2. The crescent moon shape can be modeled mathematic

1. **Problem statement:** We are given a triangle with vertices A, B, C, and a point D on the baseline from A to D. The angle at vertex B is $128^\circ$. We need to calculate the s

1. **State the problem:** We need to find the size of angle $BCD$ in the given figure. 2. **Analyze the given information:** The angle at vertex $B$ is given as $128^\circ$.

1. **State the problem:** We need to find the volume of a triangular prism with a triangular base of base length 7 m and height 4 m, and the prism length (depth) is 8 m. 2. **Formu

1. **State the problem:** Find the volume of a prism. 2. **Formula:** The volume $V$ of a prism is given by the formula:

1. **State the problem:** Find the volume of a right triangular prism with a base triangle having legs 10 cm and 4 cm, and a prism depth of 2 cm. 2. **Formula:** The volume $V$ of

1. **State the problem:** We need to find the coordinates of the vertices of triangle FGH after a 90° counterclockwise rotation around the origin. 2. **Recall the rotation formula:

1. **State the problem:** We need to find the coordinates of point $C(-3, 2)$ after it is rotated 270° clockwise around the origin.

1. **State the problem:** We need to find the coordinates of the point $R(3, 3)$ after it is rotated 180° clockwise around the origin.

1. **State the problem:** We need to find the coordinates of point $T'$, which is the image of point $T(-1, 2)$ after a 180° clockwise rotation around the origin. 2. **Formula for

1. **State the problem:** We need to find the coordinates of point $Q(4, -3)$ after it is rotated 180° clockwise around the origin. 2. **Formula for rotation:** Rotating a point $(

1. The problem asks for the central angle of the missing puzzle piece that completes a circle and a semicircle. 2. Recall that a full circle has a total central angle of $$360^\cir

1. **Problem statement:** We are given three points: $A=(5,1)$, $B=(-1,1)$, and $C=(1,4)$. We need to find a fourth point $D$ such that $A$, $B$, $C$, and $D$ form the vertices of

1. **Stating the problem:** We have a parallelogram JKLM with points J(3,3), K(4,3), L(4,1), and M(2,1). The figure is translated by the vector $$\begin{pmatrix} -6 \\ 2 \end{pmatr

1. The problem asks to identify which two angles are vertically opposite angles among the angles a, b, c, d, e, f arranged around a point. 2. Vertically opposite angles are pairs o

1. **State the problem:** Given quadrilateral ABCD with midpoints M, N, O, and P on its sides, prove that quadrilateral MNOP formed by connecting these midpoints is a parallelogram

1. **State the problem:** We have two maps showing the same forest. The first map has a scale of 1 : 20 000 and the forest area is 50 cm². The second map shows the forest area as 8

1. **Problem 1:** Find the area occupied on a plan drawn to a scale of 1 : 500 by the Olympic Stadium in Beijing, which is \(\frac{3}{5}\) of 8 (units not specified, assume 8 squar

1. **State the problem:** We have two maps showing the same forest. The first map has a scale of 1 : 20 000 and the forest area is 50 cm². The second map shows the forest area as 8

1. **State the problem:** We have triangle $\triangle OPQ$ with side $p = 9.7$ inches, angle $\angle Q = 93^\circ$, and angle $\angle O = 82^\circ$. We need to find the length of s