📐 geometry

1. **Problem statement:** Given points $X(0,0)$, $Y(7,24)$, and $Z(15,0)$, find the coordinates of point $W$ on line segment $YZ$ such that $\angle WXZ = 3 \times \angle WXY$. 2. *

1. **State the problem:** We have triangle A plotted on the coordinate plane. We need to:

1. **Problem Statement:** Given quadrilateral ABCD with (i) \(\angle DAC = \angle BCA\) and (ii) ABCD is a parallelogram, prove these properties. 2. **Step 1: Isosceles Triangle Pr

1. **Problem Statement:** We are given a regular 10-sided polygon (decagon) labeled ABCDEFGHIJ. Two angles, $x$ at vertex J and $y$ at vertex B, are marked inside the polygon. We n

1. **Stating the problem:** Given triangle ABC with points D on AC and E on AB such that BD = CE and BE = CD, we want to analyze the geometric relationships. 2. **Understanding the

1. **Problem 1:** Find the area of a triangle with two sides 29 m and 98 m, and included angle 132°. 2. **Formula:** Area = $\frac{1}{2}ab\sin(C)$ where $a$ and $b$ are sides, $C$

1. **State the problem:** We are given a triangle with angles labeled A, B, and C. The values of angles A and B are given as $121^\circ$ and $43^\circ$, respectively. We need to fi

1. **State the problem:** We have two similar L-shaped figures. The smaller one has a horizontal side of 4 mm, perimeter 28 mm, and area 18 mm². The larger one has a horizontal sid

1. **Problem statement:** We have a large square composed of 16 smaller squares, each with an area of 4 square units. We need to find the side length of the large square. 2. **Unde

1. The problem involves finding the translation vector \((m,n)\) that maps point \(X(-8,-3)\) to point \(W(0,18)\). 2. Translation moves every point by the same amount horizontally

1. **State the problem:** We need to find the volume of a cylinder with radius $r=4$ cm and height $h=9$ cm. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We need to find the volume of a cylinder with radius $r = 7$ cm and height $h = 15$ cm. 2. **Formula for the volume of a cylinder:**

1. **Талбар:** Трапецийн суурийг 3 см-ээр уртасгахад талбай 42 см²-ээр ихсэж байвал эхний өндөр $h$-г ол. 2. **Трапецийн талбайн томъёо:** $$S = \frac{a+b}{2} \times h$$

1. **State the problem:** We need to find the volume of a cylinder with diameter 10 cm and height 12 cm. 2. **Formula:** The volume $V$ of a cylinder is given by the formula:

1. **State the problem:** We need to find the volume of a cylinder with radius $r = 5$ cm and height $h = 10$ cm. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We need to find the volume of a cylinder with radius $r = 3$ cm and height $h = 8$ cm. 2. **Formula for the volume of a cylinder:** The volume $V$ is give

1. **Problem Statement:** We want to recreate the train image using equations of geometric shapes on a coordinate plane. 2. **Shapes and their equations:**

1. **State the problem:** We have two sectors OPQ and ORST with the same center O.

1. **Problem statement:** We have a right triangle with sides 18.7 mm, 16.94 mm, and 7.92 mm. The angle $x$ is between the sides 18.7 mm and 7.92 mm, and the angle opposite the 7.9

1. **Problem statement:** A circle has an area of 452 sq. cm. A chord is drawn 6 cm from the center, dividing the circle into two segments. We need to find the area of the larger s

1. **Problem Statement:** Calculate the coordinates of point $Q$ using the midpoint formula. 2. **Midpoint Formula:** The midpoint $M$ of a segment with endpoints $A(x_1, y_1)$ and