📐 geometry

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,

1. **State the problem**: We have a circle with center O, a triangle ABC inscribed in the circle, and point D on line OC produced beyond C. AD is tangent to the circle at A, and we

1. **State the problem:** We have trapezium ABCD with AD parallel to BC and \( \angle ADC = 90^\circ \). M is the midpoint of AB, \( CM = \frac{13}{2} \) cm, and \( BC + CD + DA =

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

1. **State the problem:** We have triangle $\triangle ABC$ with sides $BC=17$, $CA=18$, $AB=19$. Points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$, respectively. Point $P$

1. **State the problem:** We have triangle $\triangle ABC$ with sides $BC=17$, $CA=18$, and $AB=19$. Point $P$ is inside $\triangle ABC$ such that $PD$, $PE$, and $PF$ are perpendi

1. प्रश्न को समझें: हमारे पास दो ग्राफ हैं, जिनमें कोणों का व्यंजक दिया गया है। हमें इनमें दिए गए कोणों के मान ज्ञात करने हैं। 2. पहले ग्राफ में, चूंकि रेखाएं एक बिंदु O पर मिलती ह

1. The problem asks us to find the exact value of $x$ in a right-angled triangle with sides $3 + \sqrt{10}$, $3\sqrt{2} - \sqrt{5}$, and base $x$. The right angle is at the top ver

1. Let's restate the problem: You mention a star with 6 vertices (6 v). We want to determine what type of star this is. 2. A star polygon is typically denoted by \( \{n/k\} \), whe

1. The problem describes a shape made by attaching 6 "V" letters together to form a star-like shape. 2. Each "V" letter has 2 corners (the points where the lines meet), so 6 "V" le

1. Stating the problem: Choose the correct statement from the given options. a. A triangle has 3 sides and 4 vertices.

1. **State the problem:** We are given a rectangle ABCD with points E and F such that AD \parallel BE \parallel FC. We know the area of \triangle ABC is 1. We need to find the area

1. The problem is to find the shortest distance from the point $P(1,2,1)$ to the sphere \(x^2 + y^2 + z^2 = 3\). 2. First, calculate the distance from the center of the sphere $O(0

1. **State the problem:** We have trapezium ABCD with AD parallel to BC, \(\angle ADC = 90^\circ\), M is the midpoint of AB, \(CM = \frac{13}{2}\), and \(BC + CD + DA = 17\).

1. **Problem statement:** We have trapezium ABCD with AD parallel to BC and \(\angle ADC = 90^\circ\). M is midpoint of AB, \(CM = \frac{13}{2}\) cm, and \(BC + CD + DA = 17\) cm.

1. **State the problem:** In triangle $\triangle ABC$, the sides are given as $BC=17$, $CA=18$, and $AB=19$. Point $P$ is inside the triangle such that $PD$, $PE$, and $PF$ are per

1. **State the problem:** We have a circle with diameter AC. EC is a segment from point E (outside the circle) to C (on the circle) with length 4 cm. EF is tangent at C. Given angl

1. **State the problem:** We need to find the angle $\angle AFE$ in the given figure where $CD \parallel AF$, $\angle CDE = \angle BAF$, $AB \perp BC$, $\angle E = 80^\circ$, and $