📐 geometry

1. **State the problem:** We are given a triangle LMN with area 133 cm², side LM = 16 cm, and side MN = 21 cm. We need to find the angles \(\angle LMN\) and \(\angle MNL\). 2. **Re

1. Énonçons le problème : recalculer la distance indiquée en jaune. 2. La distance entre deux points dans un plan peut être calculée avec la formule de la distance :

1. Énonçons le problème : Calculer la distance en jaune associée à un angle de 30° dans un cercle ou une configuration géométrique donnée. 2. Formule utilisée : Pour une distance l

1. **Problem statement:** Calculate the area of the shaded portion of the triangle with two equal sides of length 8 cm and an included angle of 54°. 2. **Formula used:** The area $

1. **Problem:** Find the value of the angle given choices 150°, 120°, 75°, 60°. Since no specific question is stated, we move to the next problem.

1. **Stating the problem:** We have triangle ABC with an exterior angle bisector AD' at vertex A. We are given that segment D'C is longer than AC and segment D'B is longer than AB.

1. **Problem statement:** We have a quadrilateral with equal diagonals. The segments joining the midpoints of opposite sides have lengths 13 and 7. We need to find the area of this

1. **Problem Statement:** We need to find the fraction of the whole large square that is shaded in part (a). 2. **Understanding the figure:** The large square is divided into 4 equ

1. **Problem:** Find the equation of the circle with center $(-1, 2)$ and diameter $6$. 2. **Formula:** The equation of a circle with center $(h, k)$ and radius $r$ is:

1. **Problem statement:** We are given a parallelogram with diagonals of lengths 28 and 30, and one side length of 13. We need to find the other side length and the area of the par

1. **Problem statement:** We have a trapezoid with diagonals of lengths 6 and 10, and the segment connecting the midpoints of the bases (the mid-segment) is 4. We want to find the

1. **Stating the problem:** We need to find a sequence of rigid transformations and dilations that takes square EFGH to square ABCD.

1. The problem asks to identify sequences of transformations that show triangles ABC and AED are similar, given AC = 6 units. 2. Similar triangles can be shown by transformations s

1. The problem asks to identify the invalid step in the proof that all isosceles triangles are similar. 2. The proof attempts to show similarity by applying a sequence of transform

1. Problem: Determine if two line segments or two angles can be similar. 2. Two line segments are similar if they have the same shape and their lengths are proportional. Since line

1. **Problem Statement:** Find the length of segment $DF$ in triangle $DEF$ given:

1. დავწეროთ ამოცანა: გვაქვს წრეწირის რადიუსი $20$ სმ, ორი ურთიერთგადამკვეთი ქორდა სიგრძეებით $32$ სმ და $20\sqrt{3}$ სმ, რომლებიც ურთიერთმართობიან. უნდა ვიპოვოთ მონაკვეთები, რომლებ

1. **Problem statement:** We have a large regular pentagon with side length $x$ and diagonal length $y$. After drawing all diagonals, a smaller pentagon is formed inside. We want t

1. **State the problem:** A ladder 8 m long is leaning against a building. The bottom of the ladder is 3 m from the building. We need to find how high the ladder reaches on the bui

1. The problem is to determine if two lines \(a\) and \(b\) are coplanar, meaning they lie in the same plane. 2. Two lines are coplanar if they are either parallel, intersecting, o

1. **Problem statement:** We have a cone with apex A and a cylinder inscribed inside it. Given AG = 12 cm (height of the cone), AC = 2\sqrt{3} cm (radius of the cylinder's top circ