📐 geometry

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

1. **Problem statement:** Given triangle ABC with sides $a=7$ cm, $c=5$ cm, the incenter's distance to side BC is $1.477$ cm, and the area is $16.25$ cm². Find: 1) side $b$

1. مسئله: مقدارهای $x$ و $y$ را در دو شکل هندسی داده شده پیدا کنید. 2. برای شکل اول: دو خط موازی با یک خط قاطع داریم که زاویه‌ها $120$ و $2x + 40$ هستند.

1. **Problem Statement:** Show that the quadrilateral formed by joining BC, CD, and DA in order is a rhombus using vector methods.

1. **Problem Statement:** Given a circle with center $O$ and diameter $AB$, and points $S$ and $T$ on the circle, we need to find the measures of arcs $AST$, $SB$, $BT$, and $ATB$

1. **Stating the problem:** If a point $A(x,y)$ is enlarged at the origin by a scale factor $k$, find the coordinates of the image point $A^\prime$.

1. The problem is to understand the concept of enlargement in geometry. 2. Enlargement is a transformation that changes the size of a shape but keeps its shape and angles the same.

1. **Problem Statement:** Find the value of $y$ in the triangle configuration where $\triangle ABC$ and $\triangle DBC$ are isosceles with $AB=AC$ and $DB=DC$, $E$ is on $BC$ such

1. **State the problem:** Find the area of the given compound polygon with specified side lengths. 2. **Understand the shape:** The polygon is composed of vertical and horizontal s

1. **State the problem:** We need to find the area of a compound figure composed of two rectangles. 2. **Identify the dimensions:**

1. **State the problem:** Find the vertical distance between the points $(2, -0.8)$ and $(2, 1.2)$. 2. **Formula:** The vertical distance between two points with the same $x$-coord

1. **State the problem:** Find the vertical distance between points $A(-1, 1.5)$ and $B(-1, -1.5)$. 2. **Formula:** The vertical distance between two points with the same $x$-coord

1. **State the problem:** We have a right triangle $\triangle XYZ$ with vertices $X(1,4)$ and $Y(2,-3)$. The vertex $Z$ has positive integer coordinates and $XZ=5$. We need to find