📐 geometry

1. **State the problem:** We have a small rectangle with a height of 12 mm and an unknown width. Fourteen copies of this small rectangle are arranged in a larger rectangle composed

1. **State the problem:** We have four identical rectangles each measuring 8 cm by 3 cm arranged around a smaller purple square. We need to find the area of this purple square. 2.

1. **Problem statement:** We have two triangles PQR and RST sharing vertex R on the same baseline. Given angles \(\angle Q = 76^\circ\) and \(\angle S = 102^\circ\), sides QP = QR

1. **State the problem:** We need to find the measure of angle $\angle N$ in an isosceles triangle $\triangle NPM$ where sides $NM$ and $MP$ are equal. 2. **Identify given informat

1. **Problem statement:** Calculate the length $d$ of the cable, which is the hypotenuse $AD$ of the right triangle $ABCD$ with sides $AB=45$ m, $BC=9$ m, and $DC=28$ m. 2. **Under

1. **Énoncé du problème :** Soit un triangle rectangle en B avec les points A, B, C.

1. **Problem statement:** Calculate the perimeter of each figure given the side lengths. 2. **Formula for perimeter:** The perimeter $P$ of a polygon is the sum of the lengths of a

1. **Problem statement:** Find the coordinates of point B where the circle centered at O intersects the x-axis, and find the equation of the line on which the diameter BD lies. 2.

1. **Problem Statement:** In an acute angled triangle, AD is the median and AE is the altitude. Prove that $$C^2 = AD^2 + BC \times DE + \frac{1}{4} BC^2$$. 2. **Understanding the

1. **State the problem:** Construct a parallelogram ABCD where side AB = 6 cm, side AD = 3 cm, and the angle at vertex A is 60 degrees. 2. **Recall properties and formulas:**

1. **State the problem:** We need to find the value of $x$ given a right triangle with a vertical leg labeled $3x + 6$ and a horizontal leg labeled $30$. 2. **Understand the proble

1. **State the problem:** We need to find the value of $x$ given a right triangle where the vertical side is labeled $3x + 6$ and the horizontal side is labeled $30$. 2. **Understa

1. **State the problem:** Given a diamond-shaped quadrilateral PQRS with diagonals PR and QS intersecting at T, where PR is the perpendicular bisector of QS. Given lengths: PS = 10

1. **Problem Statement:** Calculate the value of $x$ to 4 significant figures given two circles each of radius 10.5 cm with centers $A$ and $B$ touching at $T$. Given angles $\angl

1. **Problem:** Find the unknown angle in a triangle with angles 65°, 57°, and ?. 2. **Formula:** The sum of angles in any triangle is always 180°. That is,

1. **Problem statement:** Given a triangle with sides $a$, $b$, and $c$, and the relation $$\frac{1}{a+b} + \frac{1}{b+c} = \frac{3}{a+b+c},$$ find the angle opposite side $b$. 2.

1. **State the problem:** We need to find the volume of a shape made from a hemisphere and a half-cylinder. The hemisphere has radius $r=5.3$ cm, and it sits on the half-cylinder w

1. **Problem:** The Ferris wheel has a radius of 20 m. The platform is tangent to the Ferris wheel. How far is the platform from the center of the wheel? 2. **Formula and Explanati

1. **Problem statement:** Calculate the length $x$ in the triangle where $MN \parallel BC$ and given lengths are $BM=5$, $AN=4$, and $BC=10$. 2. **Formula and theorem used:** Accor

1. **Énoncé du problème :** Montrer que le point $G$ est le barycentre des points $(D;1)$ et $(H;y)$. 2. **Rappel de la définition du barycentre :** Le barycentre $G$ des points $D

1. **Problem statement:** We want to find how a line divides a sector into two equal areas. 2. **Understanding the sector:** A sector is a portion of a circle bounded by two radii