📐 geometry

1. **Problem statement:** Given a circle with diameter CD, AE = 6 cm, and CE = 4 cm, find the radius length of the circle M. 2. **Recall the properties:** The diameter CD passes th

1. **State the problem:** Find the area of a square whose vertices lie on the circle given by the equation $$x^2 + y^2 - 4x + 6y + 4 = 0$$. 2. **Rewrite the circle equation in stan

1. **Find the unknown marked angle in each triangle.** The sum of angles in any triangle is always $180^\circ$.

1. The problem is to find the area of a rectangle. 2. The formula for the area $A$ of a rectangle is:

1. **State the problem:** We need to find the value of $y$ for the point $P = \left(-\frac{1}{7}, y\right)$ that lies on the unit circle centered at the origin. 2. **Recall the equ

1. **Problem statement:** Find the length of line segment PQ where P(-1,1) and Q(3,4). 2. **Formula used:** The distance between two points $P(x_1,y_1)$ and $Q(x_2,y_2)$ is given b

1. **Problem statement:** Find the length of line segment $PQ$ where $P(-1,1)$ and $Q(3,4)$. 2. **Formula used:** The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is giv

1. **Problem statement:** Calculate the area of trapezium ABCD, the perpendicular distance $h$ from point B to line AD, and angle DAB given:

1. **Problem statement:** Two trees stand on flat ground. The smaller tree is 7 m tall. The distance from the top of the smaller tree to the base of the taller tree is 15 m. The di

1. **Problem statement:** Find the measures of arcs and angles in a circle where given angles are 87° and 135°, and unknowns are $x$ and $y$. 2. **Relevant formulas and rules:**

1. **Problem statement:** Given the proportion $\frac{AC}{AE} = \frac{AB}{AD}$, prove that $ED \parallel CB$. 2. **Step 1: Identify the common angle.**

1. **Problem Statement:** Find the yellow area inside a square of side length 8, below the triangle formed by two radii of the inscribed circle meeting the bottom corners of the sq

1. **Problem Statement:** Given a spherical triangle with angles $A=90^\circ$, $B=100^\circ$, and $C=92^\circ$, find the side $a$ opposite angle $A$. 2. **Relevant Formula:** In sp

1. The problem asks: Which part of the spherical triangle will be encircled? 2. In spherical geometry, the term "encircled" typically refers to the part of the triangle that is enc

1. **Problem statement:** A man travels along a right-angled triangle path with base 4 km and hypotenuse 5 km. After completing the triangle, he continues 6 km in the same directio

1. **Problem statement:** ABCD is a four-sided shape but two of the sides cross. 2. **Part a:** Explain why the sum of the angles at A, B, C, and D must be less than 360°.

1. Problem: Find $x$ in a right triangle with legs 12 and 9, hypotenuse $x$. 2. Use the Pythagorean theorem: $$x^2 = 12^2 + 9^2$$

1. **Problem statement:** Ram starts at entrance E at the northernmost point of a circle with radius 50m. He walks 1/4 of the circle clockwise, then cuts across the circle to point

1. **State the problem:** We need to find the area of a sector of a circle with radius $r=5$ cm and central angle $\theta=30^\circ$. Also, complete the statement about the diameter

1. **Énoncé du problème 3 (1)** : Montrer que si A' et B' sont les symétriques par rapport à la droite XY de deux points A et B, alors les droites AB et A'B' sont égales. 2. **Défi

1. **Problem statement:** We have two right triangles sharing a side. The larger triangle has a side of 10 cm adjacent to a 45° angle. The smaller triangle has a 60° angle and an o