📐 geometry

1. Сформулюємо задачу: знайти нормальний вектор та рівняння площини, заданої рівнянням $x + 2y + 2z = 4$. 2. Формула площини у векторній формі: $Ax + By + Cz = D$, де $(A, B, C)$ —

1. Problem 15 asks: In rectangle ABCD, diagonals AC and BD intersect at O. Given \(\angle CDB = 54^\circ\), find \(\angle OAB\). 2. Important properties:

1. The problem asks to complete the congruence statement for the triangles shown in the figure. 2. Given triangles are \(\triangle YXW\) and \(\triangle ?\) with shared vertex \(W\

1. The problem asks to complete the congruence statement for the triangles in the rectangle DEFG, where \(\triangle DEG \cong \triangle ???\). 2. Given the rectangle DEFG with vert

1. The problem states that triangles \(\triangle TJM \cong \triangle PHS\). This means all corresponding angles and sides are congruent. 2. Since \(\triangle TJM \cong \triangle PH

1. **State the problem:** Given that triangles $\triangle TJM$ and $\triangle PHS$ are congruent, complete the statements about their corresponding angles and sides. 2. **Recall th

1. **State the problem:** We have a point $A(-6,3)$ and a translation rule $(x,y) \to (x+5, y-9)$. We want to find the coordinates of the image point $A'$ after applying this trans

1. **State the problem:** We need to find the perimeter of an L-shaped polygon with given side lengths: top horizontal segment = 10 cm, right vertical segment = 12 cm, and bottom h

1. Problem: In rhombus WEST, diagonals WS and TE measure 24 cm and 10 cm respectively. Find the length of side WE. 2. Formula: In a rhombus, diagonals bisect each other at right an

1. Problem II.A: Given in circle OD, $m\angle TE = 40^\circ$, find the measure of each arc UL, TY, LE, UET. 2. Since $m\angle TE = 40^\circ$ is an inscribed angle, the arc it inter

1. Problem 8: Rohit has 6 wooden sticks of equal length and wants to join them to form a regular polygon. Find the internal angle between the sticks. Formula: The internal angle of

1. **Problem Statement:** Given a circle with chord BC and a point P on BC such that AB = AP, prove that CP = CQ.

1. **Problem Statement:** We are given a triangle ABC with an inscribed circle (incircle) touching the sides at points P, Q, and R. The circle has center O and radius $r$. We need

1. **State the problem:** We need to calculate the cost of buying carpet for the living room/hall and the cost of creating a fenced lawn area, then find the total cost.

1. **Problem Statement:** Given two parallelograms ABCD and ECBF, with \(\angle A = 54^\circ\) in ABCD and \(\angle B = 66^\circ\) in ECBF, find the value of \(\angle ECF\).

1. **Problem Statement:** Given rectangle KLMN with angles \(\angle K = 47^\circ\) and \(\angle N = 57^\circ\), find the value of \(\angle KMO\) where point O lies on side KN.

1. **Problem statement:** We have rectangle PQRS with triangle TQR inside it, where TQR is isosceles with $QT = QR$, and angle $\angle PST = 26^\circ$. We need to find $\angle STR$

1. **Stating the problem:** We have trapezium ABCD with ABCE as a parallelogram and triangle ECD is isosceles with EC = ED. Given angles are \(\angle A = 105^\circ\) and \(\angle B

1. **Problem statement:** We have rectangle ABCD with sides AD = 12 and DC = 5. Points E and F lie on sides AD and DC respectively such that BE and DF are perpendicular to diagonal

1. **Problem 1: Find the diagonal $x$ of a cuboid with edges 6.3 cm, 5 cm, and 3.1 cm.** The diagonal $x$ inside a cuboid can be found using the 3D Pythagorean theorem:

1. დავიწყოთ დავწეროთ პრობლემა: გვაქვს ოთხკუთხედის გვერდები 10დმ, 15დმ, 20დმ და 25დმ. მსგავსი ოთხკუთხედის მცირე და დიდი გვერდების ჯამი არის 28დმ. უნდა ვიპოვოთ მეორე ოთხკუთხედის გვერ