📐 geometry

1. The problem is to mark a point O and draw a line from O towards the north, then draw another line from O making a bearing of 040 degrees. 2. A bearing is measured clockwise from

1. **Problem Statement:** We need to state and prove the relationship between the internal angles of a spherical triangle and the area of that triangle.

1. **Problem statement:** Define a great circle on the sphere $S^2$ and determine the number of intersection points of two distinct great circles $C_1$ and $C_2$ on $S^2$. 2. **Def

1. **State the problem:** We are given a triangle ABC inscribed in a circle with center O. Given angles:

1. **State the problem:** We are given a rectangle with width 16 cm and height 4 cm, and a square whose side length is unknown. The area of the rectangle is equal to the area of th

1. **Problem statement:** Point M is the midpoint of line segment AB. We want to understand what this means and how to find coordinates of M if coordinates of A and B are known. 2.

1. Problem: Given parallelogram MASK, find KS if MA = 9. Step 1: Recall that in a parallelogram, opposite sides are equal.

1. **Problem statement:** Find the equation and length of the altitude from vertex A in triangle ABC with vertices A(2,3), B(4,-1), and C(1,2). 2. **Recall:** The altitude from a v

1. **Problem statement:** We have two circles with centers A and B, radii 5 cm and 8 cm respectively, touching at point C on line ACB. Point D lies on the smaller circle such that

1. Problem statement: Calculate the area of a right triangle given the lengths of its legs. 2. Formula: The area $A$ of a right triangle with legs $a$ and $b$ is given by

1. **Problem:** Find $x$ in triangle ACB where segment GH is parallel to CB. Given: $AG=10$, $GH=2x+6$, $CB=25$, $CH=55$.

1. **Problem Statement:** Examine the diagonals of parallelograms by drawing a rectangle ABCD and its diagonals, measuring the diagonals, and exploring their properties. 2. **Step

1. **Stating the problem:** We have a parallelogram with two given angle measures expressed in terms of $x$: one angle is $4x - 50^\circ$ and the adjacent angle is $x + 20^\circ$.

1. The problem asks us to form an equation using the given angle measures of a parallelogram and then find the numerical values of each angle. 2. Important property: In a parallelo

1. Given: ADE, AOC, BOD are straight lines; OA = OC; OB = OD; AD = DE. 2. Triangles OAB and OCD are congruent by SAS since OA = OC, OB = OD, and angles AOC and BOD are equal.

1. **Problem statement:** Prove that quadrilateral BCED is a parallelogram given that ADE, AOC, and BOD are straight lines, OA = OC, OB = OD, and AD = DE. 2. **Recall the definitio

1. **Problem statement:** Prove that quadrilaterals ABCD and BCED are parallelograms given that lines ADE, AOC, and BOD are straight, and that $OA = OC$, $OB = OD$, and $AD = DE$.

1. **Problem statement:** We want to understand the region defined by the polar coordinate condition $|r| > 0.5$ for the angle range $\theta \in \left[-\frac{\pi}{3}, \frac{\pi}{4}

1. **Stating the problem:** Given the geometric configurations with circles, chords, and angles labeled $x$, $y$, and $z$, we need to find the values of these angles based on the g

1. **Stating the problem:** We have an L-shaped polygon with four sides: a left vertical side of 12 units, a bottom vertical side of 6 units, a right horizontal top side of 5 units

1. مسئله: مساحت مربعی را پیدا کنیم که یکی از رئوس آن نقطه $A(-1,2)$ است و یک ضلع آن روی خطی است که از نقاط $B(-1,3)$ و $C(-3,4)$ می‌گذرد. 2. ابتدا شیب خط $BC$ را محاسبه می‌کنیم با