📐 geometry

1. **State the problem:** We have a circle with center at point $P$ and diameter $\overline{AB}$. Points $A$, $B$, $C$, and $D$ lie on the circle in clockwise order. Given the angl

1. **State the problem:** We are given a circle centered at point $P$ with points $A$, $B$, and $C$ on the circle. We know the angles formed by the radii at $P$: $\angle APB = 4y +

1. Calculate the surface area of a sphere with radius 6 cm. The formula for the surface area of a sphere is $$4\pi r^2$$.

1. Calculate the surface area of a sphere of radius 6 cm. The formula for the surface area $A$ of a sphere is given by:

1. Calculate the surface area of a sphere with radius 6 cm. The formula for the surface area of a sphere is $$S = 4\pi r^2$$.

1. **State the problem:** Calculate the volume of a cone with a height (slant height provided) of 6 cm. 2. **Clarify the dimensions:** The problem gives 6 cm as the slant height (n

1. The problem is to calculate the volume of a cone with height $h=5$ cm and radius $r=2$ cm. 2. The formula for the volume of a cone is:

1. **Problem statement:** Triangle $\triangle ABC$ is inscribed in a circle with center $O$. Point $D$ lies on the extension of $OC$ beyond $C$. Line $AD$ is tangent to the circle

1. Stating the problem: We need to find the volume of a spherical bead with a diameter of 2 cm. 2. Recall the formula for the volume of a sphere: $$V = \frac{4}{3} \pi r^3$$ where

1. **Stating the problem:** We have a triangle $\triangle ABC$ inscribed in a circle with center $O$. Point $D$ lies on the extension of $OC$, and $AD$ is tangent to the circle at

1. State the problem: We need to find the radius $r$ of a spherical telescope dome given that its volume $V$ is 250 cubic meters. 2. Recall the formula for the volume of a sphere:

1. **State the problem:** We want to determine if a ball with a volume of 65.42 cm³ can fit inside a sphere with radius 3 cm. 2. **Calculate the volume of the larger sphere:** The

1. Stating the problem: We need to find the volume of air required to fill a balloon with a radius of 13 cm. 2. Recall the formula for the volume $V$ of a sphere:

1. **State the problem**: We need to find the volume of a spherical bead with diameter 2 cm. 2. **Recall relevant formula**: The volume $V$ of a sphere is given by $$V = \frac{4}{3

1. **Stating the problem:** Triangle ABC is inscribed in a circle with center O. Point D lies on the line extended from OC and AD is tangent to the circle at A. Given $\sin B = \fr

1. **State the problem:** We have a square ABCD with side length $2\sqrt{15}$. Points E and F are midpoints of sides AB and BC, respectively. 2. **Find coordinates of points:** Ass

1. **Problem statement:** Given a square ABCD with side length $2\sqrt{15}$, points E and F are midpoints of sides AB and BC respectively. Points M lies on AE, and N lies on EF, fo

1. بیایید ابتدا خط داده شده را بررسی کنیم: $$y = ax + 3a + 2$$ این خط باید از مرکز دایره عبور کند. بنابراین مرکز دایره باید نقطه اشتراک همه این خطوط باشد یعنی نقطه ای که برای همه م

1. **State the problem:** We are given triangle $ABC$ with points $D$, $E$, and $F$ as midpoints of sides $AB$, $BC$, and $CA$ respectively. We need to find the ratio of the area o

1. **بیان مسئله:** نقطه برخورد قطرهای یک مربع روی نقطه $A(2, -3)$ قرار دارد و یکی از اضلاع مربع روی خط $2x + y = 7$ منطبق است. هدف یافتن مساحت مربع است. 2. **یادآوری خصوصیات:** در

1. **State the problem:** We have trapezium $ABCD$ with $AD \parallel BC$ and right angle $\angle ADC = 90^\circ$. Point $M$ is the midpoint of $AB$, $CM = \frac{13}{2} = 6.5$ cm,