📐 geometry

1. **Problem statement:** Show that the area $A$ of an equilateral triangle with side length $x$ is given by $$A = \frac{\sqrt{3}}{4} x^2.$$\n\n2. **Recall the formula for the area

1. **State the problem:** Calculate the volume of each rectangular prism given the face area and height. 2. **Formula:** Volume of a rectangular prism is given by $$\text{Volume} =

1. **Problem:** Find the length of side JK in the right triangle JKL where angle L is 45°, side KL = 42, and JL is unknown. 2. **Given:**

1. **Problem Statement:** Given a quadrilateral with angles labeled as $d^\circ$, $c^\circ$, $(b - 10)^\circ$, and $(b + 10)^\circ$, solve for the variables $b$, $c$, and $d$.

1. **Problem:** Given a parallelogram with sides labeled $k+4$, $8$, $m$, and $11$, solve for the variables $k$ and $m$. 2. **Formula and rules:** In a parallelogram, opposite side

1. The problem states that two angles, $x^\circ$ and $71^\circ$, form a straight line. 2. Angles on a straight line add up to $180^\circ$.

1. **Problem Statement:** Given cyclic quadrilateral ABCD with tangent FG at G, chords BG and GC, and points E and F defined as intersections and extensions, prove various angle an

1. **State the problem:** Calculate the area of the shaded region inside a rectangle of length 12 m and height 7 m, with two semicircles on the left and right sides inside the rect

1. **State the problem:** Calculate the total area of a compound shape consisting of a square, an isosceles triangle on top of the square, and two semicircles on either side of the

1. **Problem Statement:** Prove the given geometric properties and find lengths in the circle with tangent AC at C, chord DF extended to A, points B and G on AC and AD such that GB

1. The problem asks to list all line segments that have the same length as segment $CH$. 2. From the graph description, segment $CH$ is marked with three short hash marks indicatin

1. **Problem statement:** We need to find the angle $v$ between two sides of lengths 3.7 cm and 4.8 cm such that the area of the triangle is 6.5 cm². 2. **Formula for the area of a

1. The problem asks to list all line segments that are the same length as segment $BG$. 2. From the graph description, segments marked with one line are equal in length. These segm

1. **Problem statement:** We have a six-pointed star made up of 6 identical quadrilaterals. Each quadrilateral has an angle $x$ inside the star and two opposite angles of $35^\circ

1. **State the problem:** We are given a triangle ABC with AB = CB, angle ABC = 50°, and the bearing of B from A is 070°. We need to find the bearing of C from A. 2. **Understand t

1. **State the problem:** We have a laser beam inside an optical fibre with parallel sides. The beam bounces off the sides at the same angle it hits them. Given the angle of incide

1. **State the problem:** We need to find the volume of a prism with a hexagonal base. 2. **Given data:**

1. **State the problem:** We need to find the volume of a prism given its cross-sectional area and length. 2. **Formula:** The volume $V$ of a prism is calculated by multiplying th

1. **State the problem:** We need to find the size of angle $q$ in a composite figure consisting of a triangle and an irregular pentagon sharing a side. 2. **Identify known angles:

1. The problem asks to find the size of angle $d$ in a regular octagon and to explain Sonia's mistake. 2. Sonia calculated $d$ as $\frac{360}{8} = 45^\circ$. This is correct for th

1. **Problem statement:** Calculate the perimeter of a right-angled triangle with legs of lengths 7 m and 16 m. 2. **Formula:** The perimeter $P$ of a triangle is the sum of the le