📐 geometry

1. **Problem statement:** Calculate the circumference, surface area, and volume of Earth modeled as a sphere with radius $r = 6.37 \times 10^6$ meters. 2. **Formulas:**

1. **Problem statement:** We have a right triangle formed by a lamp of height 3 m, a building of height $H$ meters, and the horizontal distance between them is 20 m. We need to fin

1. **Problem Statement:** We have a triangle with vertices K(-1,-2), L(-1,2), and M(2,-2). We need to find the coordinates of the vertices after a dilation centered at the origin w

1. **State the problem:** We want to find how many boxes of size 29 inches by 20 inches by 15 inches can fit on a pallet of size 40 inches by 48 inches, stacked up to 15 feet high.

1. The problem is to find the volume of a box with dimensions 29 inches by 20 inches by 15 inches. 2. The formula for the volume $V$ of a rectangular box is:

1. **State the problem:** We need to find the volume of a cylindrical pole with height $20$ m and base radius $7$ m. 2. **Formula used:** The volume $V$ of a cylinder is given by t

1. **State the problem:** We need to find the total surface area of a closed box with a base of 15m by 15m and a height of 7.5m. 2. **Formula for total surface area of a closed rec

1. Statement of the problem: The relation between the sum $y$ of the interior angles of a polygon and the number of its sides $x$ is given by the formula below. $$y = \pi(x - 2)$$.

1. **State the problem:** We are given a triangle with angles $115^\circ$, $(4k + 5)^\circ$, and $(6k + 10)^\circ$. We need to find the value of $k$. 2. **Recall the Triangle Angle

1. **Problem statement:** In the figure, ABCD is a parallelogram with the same base BC and between the same parallel lines AD and BC. (a) Write the relation between the area of tri

1. **Énoncé du problème :** On considère un triangle $ABC$ avec $I$ milieu de $[AB]$ et $J$ milieu de $[AC]$.

1. **Énoncé du problème :** Construire les points D et K définis comme barycentres des points pondérés (A ; 5) et (B ; 2).

1. **Problem Statement:** Given a rectangle PQRS and a parallelogram TQRU inside it, with RV perpendicular to TQ, we need to find the relation between the areas of PQRS and TQRU, p

1. **Énoncé du problème :** On considère un triangle ABC avec I milieu de [AB] et J milieu de [AC].

1. The problem asks to find the scale factor for the transformation of points. 2. The scale factor is found by dividing the coordinates of the image point by the corresponding coor

1. **Problem Statement:** Pentagon EFGHI is rotated 180° clockwise about point I to form pentagon E'F'G'H'I'. We need to determine which line segment is parallel to TH'.

1. **Problem Statement:** Given a circle ABCDEF with center O, tangents TA and TE, and lines BGF and COT. We know CT is parallel to BA and \(\angle OTE = 32^\circ\). We need to fin

1. **Problem Statement:** Find the volume of the solid shape composed of a square prism and a triangular pyramid on top. 2. **Given:**

1. **Problem Statement:** Find the area of triangle ABC with vertices \(A(18, -2, -12)\), \(B(14, 4, -10)\), and \(C(7, 6, -6)\). 2. **Formula:** The area of a triangle given by po

1. **Problem Statement:** We have a diagonal line crossing through 12 congruent squares arranged in two offset rows, forming a stepped shape. We need to find the value of $\cos(x)$

1. **Problem Statement:** We have a triangle with vertices A(2,6), B(2,3), and C(4,4). We want to find the coordinates of the images after reflecting the triangle first in the x-ax