📐 geometry

1. **Problem statement:** Given that line \(\overleftrightarrow{AC}\) is tangent to circle \(O\) at point \(C\), and \(\angle ACB = 28^\circ\), find the measure of \(\angle ACO\).

1. **Problem statement:** We have a circle with center $O$ and radius $OC=8$ units. The line segment $AC$ is tangent to the circle at point $C$, and $AC=15$ units. Points $A$, $B$,

1. **State the problem:** We need to find the area of a blue irregular quadrilateral with given side lengths and heights, where the base is 3 cm, a vertical height from the base to

1. The problem involves finding the measures of arcs and angles related to central angles in circles. 2. Recall that a central angle in a circle is an angle whose vertex is the cen

1. Problem: Calculate the surface area of pyramids. For a square-based pyramid:

1. **Problem statement:** Given a large circle with diameter 12 cm and a shaded annulus area of 50 cm², find the diameter of the smaller inner circle.

1. **Problem statement:** Given a logo composed of three circles each with radius $6$ cm intersecting to form a shaded equilateral triangle inside, find:

1. **Problem statement:** Given three circles I, II, and III with points T (tangency of I and III), P and Q (intersections of I and II), and PR, RS, RU lengths, find the relative p

1. **Problem statement:** We have a circle with center C and radius 6 cm. We need to find the length of the arc ADB corresponding to the central angle \(\angle CAB = \frac{\pi}{8}\

1. **Stating the problem:** We have two parallel lines cut by a transversal, creating several angles. We are asked to identify which two angles are corresponding and which two are

1. **Problem statement:** Find the area of the pathway around the rectangular pool and the area of the shaded triangle. 2. **Area of the pathway:**

1. The problem asks to identify the type of symmetry that involves a rotation isometry around a center point combined with mirror lines through the center point. 2. The key terms a

1. **Stating the problem:** We are given that \(\angle AOB = 102^\circ\), \(\angle DOE\) is 11° larger than \(\angle BOC\), and lines AC and BE are straight lines. We need to find

1. **Stating the problem:** We are given two intersecting lines at point O forming vertical angles. One angle, \(\angle POR\), is given as 146°. 2. **Understanding vertical angles:

1. **Problem Statement:** Determine the values of the missing variables and the measures of each unknown angle for the given angle relationships involving parallel lines, transvers

1. **Problem statement:** We have a cuboid with three visible faces having areas 12, 15, and 20 square meters. The edges of the cuboid are integers. We need to find the volume of t

1. **Stating the problem:** We have three squares arranged vertically between two parallel horizontal lines. The top square is rotated so that one corner touches the top line, form

1. Problem 3 involves two parallel lines cut by a transversal with angles labeled a, b, c, d and one angle given as 56°. 2. The key rule is that alternate interior angles are equal

1. **State the problem:** We have an isosceles triangle with two equal sides of length $n$ and the angle between these sides is $52^\circ$. The perpendicular height from the vertex

1. **State the problem:** We have an isosceles triangle with two equal sides of length $n$ and the angle between these sides is $52^\circ$. The perpendicular height from the vertex

1. **Problem Statement:** We need to identify the transformation that results in a horizontal reflection and a dilation by a scale factor of $\frac{1}{2}$ of an L shape. 2. **Under