📐 geometry

1. The problem asks us to find the value of $y$ such that the angle $2y + 12$ degrees is obtuse. 2. An angle is obtuse if it is greater than $90^\circ$ but less than $180^\circ$.

1. The problem asks for the order and relation between pairs of angles based on the graph and given angle measures. 2. From the figure with angles 90° and 68° and the labeled angle

1. **Problem statement:** ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. We need to show (i) triangles APB and CQD are congruent, an

1. Problem: Calculate sizes of angles in polygons and circles given various conditions. 2. (i) In a regular hexagon, central angle between two adjacent vertices (e.g., ∠AOB) is giv

1. The problem asks to solve using simple angle chasing with a solution style suitable for an IMO gold medal level in geometry. 2. Start by identifying all given angles and conside

1. **Stating the problem:** We have square $ABCD$. Point $E$ is the midpoint of $BC$. Point $F$ lies on side $AB$ such that $\angle DEF = 90^\circ$. Point $G$ lies inside the squar

1. **State the problem:** We have a parallelogram PQRS with diagonal PR. Points Q and S are joined to a point W on diagonal PR. We are given that triangles \(\Delta PQW = \Delta PS

1. **State the problem:** We are given a parallelogram PQRS with diagonal PR. Point M lies on diagonal PR, and Q and S are joined to M forming two triangles \(\triangle PEM\) and \

1. **Problem Statement:** We have an equilateral triangle $\triangle ABC$ with $AC=AB=BC$. Point $D$ is outside $\triangle ABC$ such that $AC=AD$. We need to find the value of $\an

1. **State the problem:** We have an equilateral triangle $\triangle ABC$ with points $D$ on $AB$ and $E$ on $AC$. Lines $BE$ and $CD$ intersect at $F$ such that $\angle BFC = 120^

1. Problem 23: A wall is 4m high and a ladder is placed 3m from the foot of the wall. Find the length of the ladder. - This forms a right triangle with vertical height 4m and base

1. **State the problem:** We are given the area of a square photo frame as 250 cm² and we need to find its perimeter. 2. **Recall formulas:**

1. The problem involves finding unknown side lengths (hypotenuses or legs) of right triangles using the Pythagorean theorem. 2. Recall the Pythagorean theorem: $$a^2 + b^2 = c^2$$

1. Problema 3: Calcular el largo de $AE$ dado que $AB^2 = AC^2 + BC^2$ con $AB=\sqrt{2}$ y $BE=1$. 2. Usamos el Teorema de Pitágoras para encontrar $AE^2 = AB^2 + BE^2 = 2 + 1 = 3$

1. **State the problem:** We need to construct a parallelogram ABCD where $AB = 7\text{ cm}$, $AD = 5\text{ cm}$, $\angle DAB = 65^\circ$, and $\angle ABC = 105^\circ$.\n\n2. **Und

1. The problem involves working with angles in triangles and verifying angle sums. 2. For part (a), the sum of angles in triangle ABC is given as:

1. **State the problem:** We need to find the perimeter of a combined shape consisting of a semicircle with diameter 13 m and a trapezium with parallel sides 18 m and 27 m, and hei

1. **State the problem:** We need to find the volume of a 3D shape whose cross-section is made up of a rectangle and two semicircles at its ends. The rectangle has length 10 cm and

1. **State the problem:** We need to find the total surface area of a rectangular prism with height $6$ cm, base width $8$ cm, and length $12$ cm. 2. **Recall the formula for total

1. **State the problem:** We need to find the volume of a treasure chest with a cross-section made of a rectangle and a semicircle on top. The rectangle has height 0.6 m and width

1. **Problem Statement:** You want to understand different concepts step-by-step: dot, cross, box products; conversion between Cartesian and polar coordinates; and geometric proper