📐 geometry

1. The problem is about reflecting a triangle across a line (usually the x-axis or y-axis) to obtain its mirror image. 2. To reflect a triangle over the x-axis, take each vertex wi

1. The first problem asks to find the size of all angles in a right triangle where one angle is $90^\circ$ and the other two angles are unknown. 2. Since the triangle is right-angl

1. The problem is to find the missing angles in several triangles, each with a 90° right angle, using properties of angles in a triangle, straight lines, and at a point. 2. We know

1. The problem states that the area of a circle is given as 20 and the diameter is given as 10, which implies the radius $r$ is 5 because $r = \frac{diameter}{2} = \frac{10}{2} = 5

1. Let's clarify the problem: We have a rectangle with one side marked by 1 tick (shorter side) and the other by 2 ticks (longer side). 2. Assuming the shorter side length is $x$ u

1. Let's clarify the problem statement: We are working with a rectangle where the longer side is marked with double 2 tick marks, while the shorter side has a single 2 tick mark. 2

1. **State the problem:** We need to find the reflection of triangle ABC over the vertical line \(\ell\) given by \(x=1\). The vertices of the original triangle are \(A=(3,-4)\), \

1. **Problem Statement:** We need to understand the position of the corner relative to the diameter. It is given that the corner touches the diameter but is not at its midpoint. 2.

1. **State the problem:** We have a semicircle with diameter 10 and a rectangle inscribed inside it. One corner of the rectangle is at the midpoint of the diameter, and the opposit

1. **Identify circle parts in \( \emptyset O \):**\n\n- Radii: segments from center \( O \) to circle points, e.g., \( OJ \) and \( OS \).\n- Diameters: chords through \( O \) pass

1. First, let's restate the problem in English:\nWe need to find the minimum length of iron rod needed to make 100 window grilles shaped as circular sectors (juring) with radius $1

1. The problem describes a coordinate system with an angle $\theta$ formed at the origin between the $x$-axis and a ray extending upward right. 2. Points $N$, $M$, and $M'$ lie on

1. The problem describes two coordinate systems sharing the same origin $O$, with axes $X, Y$ and rotated axes $X', Y'$. 2. The angle between the original $X$ axis and the rotated

1. The problem involves finding the equation of the line of reflection (mirror line) for triangle POR reflected to triangle P'Q'R'. 2. Given points are P(6,2), R(9,2) in the origin

1. **Stating the problem:** We are given a triangle PQR with angles \(x^\circ\), \((x + 30)^\circ\), and \(3x^\circ\). We need to find the value of \(x\).\n\n2. **Using the triangl

1. **Stating the problem:** We have a right triangle where the vertical side is 6 units, and the horizontal side is given as 10 feet 8 inches. The angle adjacent to the horizontal

1. **State the problem:** Determine which sequences of transformations prove congruency by mapping polygon I onto polygon II. 2. **Analyze Shape Locations:** Polygon I is in quadra

1. The problem involves determining the values of angles \(\angle 1\) through \(\angle 27\) in a circle with many chords and points. 2. Given the complexity and the provided formul

1. **Problem statement:** Points A and B lie on the circumference of a circle with center O and radius $r$. The angle $\angle AOB = \theta$ radians and $C$ is the midpoint of $OA$.

1. The angle formed by a tangent and a secant outside a circle is half the difference of the intercepted arcs. Here, arcs are 200° and 220°. $$x=\frac{|220-200|}{2}=\frac{20}{2}=10

1. **Problem Statement:** We analyze Diagram B and solve the questions: identify triangles, check for congruency, complete the transformation table, and discuss similarity and cong