📐 geometry

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

1. **Problem statement:** We have two circles: a smaller circle with center A and radius $r$ passing through points B, C, and D, and a larger circle with center C and radius $s$ pa

1. დავწეროთ მოცემული პირობა: ორი მსგავსი სამკუთხედის პერიმეტრები შეეფარდება როგორც $10:9$. 2. პირველი სამკუთხედის გვერდების შეფარდება არის $6:7:8$.

1. **Problem Statement:** Find the midpoint of a line segment given endpoints, and find the other endpoint given one endpoint and the midpoint.

1. დავწეროთ მოცემული პირობა: ორი მსგავსი სამკუთხედის პერიმეტრები შეეფარდება ერთმანეთს როგორც $10:9$. 2. პირველი სამკუთხედის გვერდების შეფარდება არის $6:7:8$.

1. **Problem Statement:** Calculate the area of the shaded part formed by the intersection of the given shapes inside the square of side length 16.

1. **Problem statement:** (a) Show that $s = \sqrt{3} r$ given the two circles and points described.

1. **Problem statement:** We have two circles: one with center A and radius $r$ passing through points B, C, and D, and a larger circle with center C and radius $s$ passing through

1. **State the problem:** We have a circle with diameter endpoints P(1,1) and Q(7,11).

1. The problem asks us to find the area of the shaded part. 2. To find the area of a shaded region, we typically need to know the boundaries of the region, such as curves, lines, o

1. **Problem statement:** (a) Show that $s = \sqrt{3} r$ where $s$ is the radius of the larger circle centered at $C$ passing through $B$ and $D$, and $r$ is the radius of the smal

1. **Problem Statement:** We have a cuboid PQRSTUVW with edges:

1. **Problem Statement:** We have a cuboid PQRSTUVW with dimensions: