📐 geometry

1. The problem is to find the final coordinates of a point after rotation on a graph. 2. The formula for rotating a point $(x,y)$ by an angle $\theta$ counterclockwise about the or

1. **State the problem:** We have a triangle with vertices A(0, 2), B(-10, 1), and C(-9, 6). We need to rotate this triangle 270° clockwise about the origin. 2. **Formula for rotat

1. **Problem statement:** We have a circle with center at point $O(0,0)$ and a point $N(1,0)$ on the circle. Point $M(a,b)$ is also on the circle. The angle $\angle NOM$ measures $

1. **Problem statement:** Given triangle ABC with points A(8,0), B(0,0), and angle $\angle BAC = 300^\circ$, two points start moving simultaneously: one from A towards C along AC a

1. **State the problem:** We have a large rectangle divided into 20 small squares (5 rows × 4 columns). Each small square has an area of $9\text{ cm}^2$. Among these, 11 squares ar

1. **State the problem:** We have four intersecting lines forming two pairs of parallel lines cut by two transversals, creating angles 49° and 58° at the top left, and unknown angl

1. **State the problem:** We have two intersecting lines forming angles 120°, $x$, and 125°. Angle $x$ is opposite the 120° angle, and angle $y$ is adjacent to the 125° angle. We n

1. **Problem:** Find the volume of a cylinder with radius 6 cm and height 5 cm, using $\pi = 3.14$. 2. **Formula:** The volume $V$ of a cylinder is given by

1. **Stating the problem:** We are given points of a quadrilateral with vertices \(A(1,3), B(4,1), C(6,4), D(3,6)\) and vectors \(\vec{u} = \overrightarrow{AB} = (3, -2)\) and \(\v

1. **Problem Statement:** Given that angle 1 is congruent to angle 8 ($\angle 1 = \angle 8$), determine which lines are parallel. 2. **Understanding the Setup:** Lines $l$ and $m$

1. **Problem statement:** Given that the pairs of angles \(\angle 13 = \angle 2\), determine which lines are parallel. 2. **Understanding the problem:** When two lines are cut by a

1. The problem seems to involve understanding the relationship between the hypotenuse (hp) and the side length of a right triangle. 2. In a right triangle, the hypotenuse is the lo

1. **State the problem:** We need to find the value of $x$ such that point $N$ is the incenter of triangle $QRP$. The incenter is the point where the angle bisectors intersect and

1. **Problem statement:** When the given net is folded into a cuboid, determine which two points join with point X. 2. **Understanding the net:** The net consists of six rectangles

1. The problem is to find the surface area of a triangular prism given its net on a 1 cm² grid. 2. The net consists of two triangles and three rectangles. The surface area of the p

1. **State the problem:** Rotate the polygon with vertices A(0, -3), B(-6, 0), C(-7, -10), and D(-3, -9) by 270° clockwise around the origin. 2. **Formula for rotation:** Rotating

1. **State the problem:** We need to rotate the shape with vertices A(1,3), B(8,0), C(9,8), and D(2,7) by 90° clockwise around the origin on the coordinate plane. 2. **Formula for

1. The problem is to rotate a shape 90 degrees clockwise around the origin in the coordinate plane. 2. The formula for rotating a point $(x,y)$ 90 degrees clockwise is:

1. **State the problem:** We have a polygon with vertices at points A(-3, 5), B(-7, 2), C(-8, 8), and D(-4, 10). We need to rotate this shape 90° clockwise around the origin. 2. **

1. **State the problem:** We have a shape with points A(-4,0), B(-7,-2), and C(-9,-9). We need to rotate this shape 180° counterclockwise about the origin. 2. **Formula for rotatio

1. **State the problem:** We have two vertical poles, one 16 ft high and the other 24 ft high, standing 30 ft apart. A worker wants to place a stake on the ground between them and