📐 geometry

1. **Problem statement:** We have two circles touching at point $C$. Points $A$, $B$, and $C$ lie on the smaller circle with center $O$. Points $C$, $D$, $E$, and $F$ lie on the la

1. **Problem Statement:** Calculate the following for the triangular prism with given lengths:

1. **Problem statement:** Find $x$ in the given figure where two parallel horizontal lines are cut by a transversal, with angles $x^\circ$ and $70^\circ$ as shown. 2. **Relevant ru

1. **State the problem:** We need to find the area of the polygon with vertices at points $$(0,5), (7,9), (15,2), (19,8)$$ above the x-axis. 2. **Formula used:** The area of a poly

1. **Problem Statement:** We want to find the approximate length of the wire $AD$ that crosses the canyon from point $A$ to point $D$.

1. **State the problem:** We need to find the measure of $\angle E$ in triangle $DEF$ where $\angle D = 85^\circ$, side $DF = 9$ mm, and side $EF = 16$ m. 2. **Important note:** Th

1. **Problem statement:** Find the length of side $AB$ in triangle $ABC$ where angle $A = 52^\circ$, angle $C = 72^\circ$, and side $AC = 500$ m. 2. **Step 1: Find angle $B$.**

1. **Problem:** Construct an irregular polygon ABCDE with given sides and angles, then measure the length of AE. 2. **Step 1: Understand the polygon construction**

1. **State the problem:** We need to find the length of the fence around a circular pool excluding the gate width. 2. **Given data:**

1. **State the problem:** We need to find the missing side length $k$ of a rectangle where the perimeter $P$ is $20$ meters and one side length is $3$ meters. 2. **Recall the formu

1. **State the problem:** We need to find the missing side length $d$ of a rectangle given its perimeter $P = 18$ cm and one side length $3$ cm. 2. **Formula for perimeter of a rec

1. **State the problem:** We need to find the exact area of the shaded annulus (ring) formed between two concentric circles with radii $22$ ft and $29$ ft. 2. **Formula used:** The

1. **State the problem:** We need to find the area of a sector of a circle with radius $5$ meters and central angle $243^\circ$. 2. **Formula for the area of a sector:** The area $

1. **State the problem:** Identify which two angles form a linear pair in the given diagram. 2. **Recall the definition:** A linear pair consists of two adjacent angles whose non-c

1. **State the problem:** We are given a right triangle with angles including $a = 23^\circ$ and need to find $c^\circ$, the angle the pole makes with the athlete's body. 2. **Unde

1. **Problem Statement:** We have a rectangle inside a quadrant (a quarter circle). The rectangle has perimeter $46$ cm and area $76$ cm$^2$. The length $PQ$ (top side) is $20$ cm.

1. **Problem Statement:** Given $AD=100$ cm, $CD=64$ cm, and point $D$ lies on line segment $AC$. Triangles $\triangle ABD$ and $\triangle BCD$ are similar. We are asked to find (a

1. **Problem Statement:** We need to find which expression is equivalent to $m\angle 4$ based on the given angles in the triangle-like figure. 2. **Understanding the figure and ang

1. **Problem Statement:** Determine which statements about the angles of the given triangle and its exterior angles are true. 2. **Key Definitions:**

1. **Problem Statement:** Given points A, B, D on a circle with CD tangent at D, AED straight line, E on AD such that BE \parallel CD, AB = BD = CD, BC = y units, AB = 2 units, and

1. **Problem statement:** We have an isosceles triangle with two equal sides of length 14 cm and a base of 24 cm. We need to find the length $x$ of the altitude (height) drawn from