📐 geometry

1. **State the problem:** We have a circle with center $p$ and an arc $qr$ measuring 144 degrees. We need to find the degree measure of angle $pqr$. 2. **Recall the relevant rule:*

1. **State the problem:** Calculate the perimeter of quadrilateral ABCD, which is circumscribed around a circle. 2. **Important property:** For a quadrilateral circumscribed around

1. **State the problem:** We have two similar right circular cylinders A and B. Cylinder A has volume $112\pi$ and radius 4 units.

1. **State the problem:** We have two similar right rectangular prisms X and Y. - Surface area of X: $58$ cm²

1. Problem statement: Calculate the volume and surface area of a prism with a triangular base with sides 3 cm, 4 cm, 5 cm and height 5 cm. 2. Formula for volume of a prism: $$V = A

1. **Problem statement:** Prove that point A is the orthocenter of triangle HAC (d) and that segment AK equals segment AD (f).

1. **Problem Statement:** In right triangle $PQR$ with right angle at $Q$, segment $QT$ is perpendicular to hypotenuse $PR$. We need to determine which of the given statements must

1. **Problem statement:** Given triangle $\triangle ABC$ with $AB=9$cm, $BC=12$cm, and $AC=8$cm, triangle $\triangle AB'C'$ is obtained by rotating $\triangle ABC$ about point $A$

1. The problem asks for the smallest angle of rotation that maps the image onto itself, given it has rotational symmetry. 2. The image is described as an 8-part pinwheel with alter

1. The problem: Explain the triangle angle sum rule. 2. The triangle angle sum rule states that the sum of the interior angles of any triangle is always equal to 180 degrees.

1. **Problem statement:** Given the lines $y = -x + 5$ and $y = x + 1$, which are perpendicular, identify all similar triangles formed by these lines and the coordinate axes, and j

1. **Problem Statement:** We have a geometric figure with segments labeled 9, 5, 7, 20, and an unknown length $x$. We need to find the value of $x$. 2. **Understanding the figure:*

1. The problem asks to find the missing angle $h$ in a nonagon (9-sided polygon) where the other 8 angles are given as 164°, 135°, 150°, 113.3°, 160°, 135°, 163°, and 100.2°. 2. Th

1. The problem asks for the sum of the interior angle measures of a heptagon. 2. A heptagon is a polygon with 7 sides.

1. **Problem Statement:** An artist designs a logo using a regular hexagon with identical star icons at each vertex. We need to find how many times the logo appears unchanged when

1. **Problem Statement:** A quadcopter has 4 identical arms spaced 90 degrees apart. We need to find the order of rotational symmetry of its body and determine if a 270-degree cloc

1. The problem asks to find at least four capital letters from the English alphabet that have rotational symmetry of order 2. 2. Rotational symmetry of order 2 means the letter loo

1. **State the problem:** We have a Ferris wheel with 8 evenly spaced gondolas arranged in a circle. We need to find the order of rotational symmetry and the smallest angle of rota

1. **State the problem:** A particle moves on a unit circle centered at the origin, starting at point $(1,0)$. It undergoes a rotation corresponding to a shape with rotational symm

1. **Problem statement:** A square centered at the origin with corners at (2, 2), (-2, 2), (-2, -2), and (2, -2) is rotated 90 degrees clockwise around the origin. Find the new coo

1. **Problem:** Rotate the square centered at the origin with corners at (2, 2), (-2, 2), (-2, -2), and (2, -2) by 90 degrees clockwise around the origin. Find the new coordinates