📐 geometry

1. **Problem Statement:** We are given a circle with chords AB and CD. AB is 16 units long. CD is divided into two segments labeled $x$ and $2x$. The segment between points A and C

1. **Problem Statement:** We are given a circle with center C and points A, B, D, E, F on the circle. Inside the circle, triangle ABC has an angle at A measuring 109°.

1. **Stating the problem:** We are given a polygon with sides labeled $a$, $b$, and $c$, all greater than zero. We need to determine which of the given expressions is *not* equal t

1. Problem: Find the measures of angles $\alpha$, $\beta$, $\gamma$, and $\delta$ given the values: $\alpha = 138^\circ$

1. **Problem statement:** Given a circle with chords SN and MP intersecting at Q, tangent SR at S, and angles $\angle SRP=26^\circ$, $\angle QMS=52^\circ$, $\angle PQS=76^\circ$, f

1. The problem asks us to identify which two shapes are similar. 2. Similar shapes have the same shape but may differ in size. This means their corresponding angles are equal and t

1. **Problem statement:** Given points A(0,1), B(3,4), and C(1,3), we need to show that $\cos \angle ACB = -\frac{4}{5}$ and calculate the area of triangle $\triangle ABC$. 2. **To

1. **Problem:** Find the new equation of the line after a 90° clockwise rotation about the point (-2, 3) for the line $y=2x+2$. 2. **Formula and rules:**

1. The problem is to understand and apply the triangle postulates to determine if two triangles are congruent. 2. The main triangle postulates are:

1. **Stating the problem:** We need to find the size of angle $e$ given that it is adjacent to a $78^\circ$ angle and there is a right angle ($90^\circ$) formed by a vertical dashe

1. The problem involves finding the measures of various angles and segments in multiple circles with given points and expressions. 2. We start with the first set: m\angle MKN, m\an

1. **Problem statement:** Find the length of side JK in each triangle given angles and sides. 2. **Key formulas and rules:**

1. **Problem Statement:** Find the length of side JK in each triangle given angles and some side lengths. 2. **Key Formula:** Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\s

1. **State the problem:** We are given two angles, $4p$ and $5p$, that are adjacent and together form a right angle (90 degrees). We need to find the value of $p$. 2. **Formula and

1. The problem states that two angles, labeled as $3p$ degrees and $7p$ degrees, form a right angle at their intersection. 2. We know that the sum of angles forming a right angle i

1. **Problem statement:** We need to find the distance between points A(2,3) and B(6,7), and between points C(-4,2) and D(0,-2) on a Cartesian plane. 2. **Formula used:** The dista

1. **Problem Statement:** Given a circle with radius $r=5$ ft and a chord $AB$ whose perpendicular distance from the center $O$ is $d=3$ ft. Points $A$ and $B$ lie on the circle.

1. **Problem Statement:** Find the coordinates of the images of vertices B, C, and D of triangle △BCD under a dilation centered at the origin with scale factor 3.

1. **Problem Statement:** We have a large square, and inside it, a smaller square is inscribed by connecting the midpoints of the sides of the larger square. This process is repeat

1. समस्या: त्रिभुज 𝐴𝐵𝐶 र 𝑃𝑂𝑅𝑆 का सममिति केन्द्रहरू के हुन् र कसरी तिनीहरूलाई विस्तार गर्न सकिन्छ? 2. सममिति केन्द्र (Symmetry center) भनेको कुनै आकृतिको यस्तो बिन्दु हो जसको वरिपरि

1. समस्या: त्रिभुज ABC का शीर्षविन्दुहरू A(4,-2), B(3,1), र C(2,5) लाई विस्तार गुणांक $E=[0,2]$ द्वारा विस्तार गर्नुहोस्। 2. विस्तारको सूत्र: यदि कुनै बिन्दु $P(x,y)$ लाई विस्तार ग