📐 geometry

1. **Problem statement:** We have two containers: one cylindrical and one spherical. Both have the same volume. The cylinder has a height of 50 cm and a radius of 11 cm. We need to

1. **Problem statement:** We have two containers: one cylindrical and one spherical. Both have the same volume. The cylinder has a height of 50 cm and a radius of 11 cm. We need to

1. **Problem Statement:** Given a parallelogram ABFD with points D, A, B, and C on a circle, and lines BF and DF extended to meet DC and CB at points E and G respectively, with DC

1. **Problem Statement:** Given points W, X, Y, Z and lines WX, WY, WZ, XY, XZ, YZ, prove the theorem: "If W is a point, there exists at least one line not containing W." 2. **Unde

1. **Problem Statement:** Given points W, X, Y, Z and lines formed between pairs (W,X), (W,Y), (W,Z), (X,Y), (X,Z), and (Y,Z), prove the theorem: "If W is a point, there exists at

1. **Problem statement:** Given points A(-2, 5), B(4, 3), and O(0, 0), find the equation of line AB, length of AB, perpendicular distance from O to AB, area of parallelogram OABC,

1. **Problem Statement:** We have a children's park consisting of a quadrilateral ABCD and a semicircle AED. Given side lengths AB = 24 m, BC = 7 m, CD = 25 m, and the semicircle w

1. **Problem Statement:** We are given a circle with points A (top), B (bottom), P (left), Q (right), R (bottom-left), S (bottom-right), and center O. Points M and N lie on the ver

1. The problem is to find the formula for the center and radius of a circle given its equation. 2. The general form of a circle's equation is $$ (x - h)^2 + (y - k)^2 = r^2 $$ wher

1. **State the problem:** Find the surface area of a staircase with three steps, where the total length is 90 cm, the depth of each step is 25 cm, and the total height is 16 cm. 2.

1. **State the problem:** Find the surface area of a set of three steps with dimensions: height of smallest step = 16 cm, depth of top step = 25 cm, total depth = 90 cm. 2. **Under

1. **Stating the problem:** We need to find the sizes of the unknown angles $a$, $b$, $c$, $d$, and $e$ in the given triangle diagram, where some angles and parallel lines are give

1. **Problem Statement:** We have a semicircle centered at point $O$ with radius 8 cm. Points $A$ and $B$ are endpoints of the diameter. Point $C$ lies on the semicircle. We need t

1. **Problem Statement:** Fill in the blanks and calculate areas of given shapes including rectangles, polygons, parallelograms, and triangles.

1. **State the problem:** We are given a triangle $\triangle ABC$ with angles $A = 29^\circ$, $B = 36^\circ$, and side $b = 15.8$ cm. We need to find the length of side $a$ opposit

1. The problem is to find the supplementary angle of 135 degrees. 2. Supplementary angles are two angles whose measures add up to 180 degrees.

1. **Problem Statement:** Given a circle with center O and points P, Q, R, S on the circle such that PQ = SR = 10 cm, and a triangle ABC with \(\angle ACB = 90^\circ\) intersecting

1. **Problem statement:** Two angles are supplementary, and one angle measures 45 degrees. Find the measure of the other angle. 2. **Formula and rule:** Supplementary angles add up

1. **Problem Statement:** We are given a quadrilateral ABCD with diagonals intersecting at point E such that $AE = EC$ and $BE = ED$. We need to determine if ABCD is a parallelogra

1. **Problem Statement:** We need to find the coordinates of points D, E, and F after a 90° clockwise rotation around the origin. 2. **Rotation Formula:** For a point $(x,y)$, a 90

1. **Problem Statement:** We have two circles intersecting at point A. AB is tangent to the larger circle at A, AC is tangent to the smaller circle at A, and AD is a common chord.