📐 geometry

1. **Problem statement:** We have a polygonal side view of a building with given side lengths and angles: $v=10^\circ$, $u=40^\circ$, and we need to argue that $w=40^\circ$, then f

1. **Problem statement:** Calculate the perimeter of the shaded region and the volume of wood to be cut off from the wooden door stopper. 2. **Given:**

1. **Stating the problem:** We have a sequence of right triangles where the shortest side $a=1$ is fixed. The triangles are:

1. **Stating the problem:** Calculate the area ($a$) and perimeter ($p$) of the right triangle in the middle-right position with sides 6 cm, 8 cm, and hypotenuse 10 cm.

1. **State the problem:** We need to find the volume of a shape made by combining a cone and a hemisphere. The hemisphere has radius $r=7$ cm, and the cone has radius $r=7$ cm and

1. **Énoncé du problème :** On a les points $A(-2;3)$, $B(4;1)$, et $C(3;-2)$.

1. **Problem statement:** We are given a triangle ABC with sides AC = $7\sqrt{3}$ cm, AB = $\sqrt{6}$ cm, BC = $a$ cm, and angle $\angle A = 45^\circ$. We want to show that $a = \s

1. **State the problem:** We need to find the coordinates of the image R''S''T''U'' after reflecting the figure RSTU across the x-axis. 2. **Recall the reflection rule across the x

1. **State the problem:** We are given a triangle \(\triangle DEF\) with sides \(f = 830\) cm, \(d = 520\) cm, and angle \(\angle E = 129^\circ\). We need to find the length of sid

1. **State the problem:** Given that CA bisects \(\angle BAD\), \(AB \perp BC\), and \(AD \perp DC\), prove that \(\triangle ABC \cong \triangle ADC\). 2. **Identify given informat

1. **State the problem:** Given quadrilateral ABCD with AB \parallel CD and AC bisects BD, prove that BD bisects AC. 2. **Understand the given information:**

1. **Stating the problem:** We have a circle with center M and radius $r = 6370$ km. There is a sector formed by two radii with a central angle of $142^\circ$. We want to find the

1. **Problem Statement:** Find the area of the large semicircle given smaller semicircle and quarter circle inside it, each labeled with radius 4. 2. **Understanding the problem:**

1. **State the problem:** We want to calculate the area of a square. 2. **Formula:** The area $A$ of a square is given by the formula:

1. The problem involves understanding a circular arc centered at the fraction $\frac{1}{2}$ with a radius given by the fraction $\frac{2}{4}$.\n\n2. The center of the circle is at

1. **Problem Statement:** We have a cube net with faces labeled E, A, B, C, D, F arranged as shown:

1. **Stating the problem:** We are given a right triangle with sides 20 and 25, and a side $x$ opposite the right angle. We want to find the value of $x$.

1. Problem: Find the volume of a composite solid consisting of a hemisphere on top of a cube, each side of the cube is 8 cm. 2. Formulae:

1. **Problem statement:** We have two circles with centers O and P. Line segment RT is tangent to the larger circle at R and to the smaller circle at S. Given lengths are $OR=9$ m,

1. **Problem Statement:** Find the surface area of a solid formed by joining four cubes each with side length 25 cm in two different arrangements.

1. **Problem statement:** Given a square ABCD with side length 8, a smaller square is formed by joining the midpoints of each side of ABCD. We need to find the perimeter of this sm