📐 geometry

1. **Problem statement:** We have an equilateral triangle ABC and a point P inside it. From P, three perpendiculars are dropped to the sides BC, AC, and AB, meeting at points D, E,

1. **State the problem:** Write a two-column proof for a given geometric statement or theorem. 2. **Understand the two-column proof format:** It consists of two columns: the left c

1. **Problem statement:** Given that BE and AD bisect each other, prove that segment AB is congruent to segment DE. 2. **Understanding the problem:** When two segments bisect each

1. **Problem statement:** We need to find the total length of the highway going around the bay, which consists of two segments: one 7 miles long at a 145° angle and another 2 miles

1. **Stating the problem:** Given points A(3, -2) and B(9, 11), find the direction of translation from point A to point B.

1. **State the problem:** We need to find the volume and surface area of a triangular prism with sides of the triangular base $s_1=3.75$, $s_2=5.5$, $s_3=2.25$, length $L=22$, and

1. **State the problem:** We need to find the volume and surface area of a triangular prism with side lengths of the triangular base $s_1=3.75$, $s_2=5.5$, $s_3=2.25$, length $L=22

1. **Problem statement:** Given $mCG = 6x + 5$, $mAR = 4x + 15$, and $m\angle AEC = 120$, find: a) $x$

1. **State the problem:** Find the area of the L-shaped composite figure made of two rectangles with dimensions 10 cm by 6 cm and 8 cm by 12 cm. 2. **Formula:** Area of a rectangle

1. **State the problem:** We have a right circular cone where the slant length $l$ is equal to the diameter of its base. We need to find the volume of this cone. 2. **Known formula

1. **State the problem:** We have an isosceles right triangle with angles 45°, 45°, and 90°. We want to explore the relationship between the lengths of the legs and the hypotenuse.

1. **State the problem:** We have a 45°-45°-90° triangle with angles $m\angle ABC = 45^\circ$ and $m\angle ACB = 45^\circ$. We want to find the lengths of the legs $AB$, $AC$, and

1. Задача: Ребро куба зменшили в 3 рази. У скільки разів зменшився його об’єм? Формула об’єму куба: $$V = a^3$$, де $a$ — довжина ребра.

1. **Problem Statement:** Given parallelogram PURE with diagonals PR and EU intersecting at O. We know diagonals of a parallelogram bisect each other, so $O$ is the midpoint of bot

1. Задача: Знайти об’єм призми, якщо площа основи $S=12$ см², а висота $h=5$ см. Формула об’єму призми: $$V = S \times h$$

1. **Problem statement:** We have an isosceles triangle ABC with AB = AC = 13 cm and BC = 10 cm lying in a horizontal plane.

1. Задача: Знайти об’єм призми, якщо площа основи $S=12$ см², а висота $h=5$ см. Формула об’єму призми: $$V = S \times h$$

1. **Problem Statement:** We are given three triangles: \(\triangle ABC\), \(\triangle DEF\), and \(\triangle IGH\) with side lengths:

1. The problem involves comparing three triangles: \(\triangle ABC\), \(\triangle DEF\), and \(\triangle GHI\) with given side lengths. 2. We want to determine which triangle is si

1. The problem asks: "Where is the shaded area?" This usually refers to identifying the region on a graph or figure that is highlighted or filled. 2. To determine the shaded area,

1. The problem involves solving a triangle using the SSS (Side-Side-Side) theorem and the TABT (The Angle Bisector Theorem). 2. The SSS theorem states that if three sides of one tr