📐 geometry

1. **Problem statement:** We need to find the radius $r$ of the circular arc and then determine the length of the arc from point $E$ to point $F$.

1. Staðfesta verkefnið: Finna samanlagt flatarmál skyggðu svæðanna sem samanstanda af tveimur rétthyrndum þríhyrningum og hring með gefnum stærðum. 2. Gefnar stærðir eru:

1. **Problem statement:** Jane wants to find the length of the hypotenuse $|AC|$ of the right triangle $ABC$ where $AB=2$ m and $BC=3$ m, and then find the angle $\theta$ at $C$.

1. **Problem Statement:** We have a circle with points A, B, D, and E on its circumference and center O. EB is a diameter, BC is tangent at B, angle EBD = 30°, and angle ACB = 20°.

1. **Stating the problem:** We need to calculate the area and perimeter of composite figures made up of rectangles, triangles, and semicircles. 2. **Formulas used:**

1. **State the problem:** We need to find the area of a circular pond with a radius of 3.67 meters. 2. **Formula:** The area $A$ of a circle is given by the formula:

1. **State the problem:** We have a rectangle with length $127.3$ cm and width $86.5$ cm, both measured to 1 decimal place. We need to find the upper and lower bounds for the perim

1. The problem is to find the remaining interior angle of a polygon when the sum of the interior angles is 540 degrees. 2. The formula for the sum of interior angles of a polygon w

1. **State the problem:** We have a triangle sharing a side with an irregular pentagon. We know four interior angles of the pentagon: 61°, 113°, 22°, and 108°. We want to find the

1. **State the problem:** We need to find the radius $r$ of a cylindrical glass that holds 150 ml (which equals 150 cm³) of juice up to a height of 8 cm. 2. **Formula used:** The v

1. **State the problem:** We have a cylindrical glass filled with juice to a height of 8 cm, and the volume of juice is 150 cm³. We need to find the radius $r$ of the glass.

1. **State the problem:** Calculate the shaded area in the semicircle diagram where the radius $r=25$ cm and $OM=18$ cm. 2. **Identify the shaded area:** The shaded area is the seg

1. **State the problem:** We have a semicircle with center O and radius 25 cm. M is the midpoint of chord AB, and OM = 18 cm. We need to (a) show that angle AOB = 87.9° correct to

1. **State the problem:** We need to find the total surface area of a cylinder with height $h=7$ m and radius $r=4$ m. 2. **Formula for total surface area of a cylinder:**

1. **State the problem:** We have a cylinder net consisting of two circles each with radius 5 mm and a rectangle with height 3 mm and length $x$.

1. **State the problem:** We have two triangles, ONM and KML, with NO parallel to KL. Given lengths are KL = 21, NO = 14, ML = 24, and KM = 15. We need to find the length of NM. 2.

1. **State the problem:** We have an L-shaped polygon with given side lengths 4 cm, 6 cm, and 10 cm, and an unknown top horizontal side labeled as [Box] cm. The total perimeter is

1. **Problem statement:** We have quadrilateral ABCD with given sides |AD| = 4 cm, |DC| = 6 cm, and angles \(\angle ADC = 60^\circ\), \(\angle CAB = 30^\circ\), and \(\angle ABC =

1. **Problem statement:** Given a prism ABC.A'B'C' with M as the midpoint of BB'. Given vectors \(\overrightarrow{AC} = \vec{a}, \overrightarrow{BC} = \vec{b}, \overrightarrow{AA'}

1. **State the problem:** We have two parallel lines cut by a transversal, creating angles of 72° and y° as shown. 2. **Identify the relationship:** When two parallel lines are cut

1. **Calculate the area of each shape** (i) L-shaped polygon made of rectangles: