📐 geometry

1. **Problem:** We need to find the coordinates of the point where the two circles touch. The circles are given by the equations $$((x+7)^2)+((y+2)^2)=4$$ and $$((x+7)^2)+((y+5)^2)

1. **State the problem:** Determine which given transformations carry a regular polygon (an equilateral triangle in this case) onto itself. 2. **Analyze each transformation:**

1. The question asks if the described shape has rotational symmetry. Rotational symmetry occurs when a figure looks the same after being rotated less than a full turn (360 degrees)

1. Problema 20: Avem triunghiul isoscel ABC cu AB \equiv AC. M și N sunt mijloacele laturilor AB și AC, iar D este mijlocul laturii BC. P și Q sunt piciorul perpendicularelor din D

1. **Problem:** Circle A has a circumference of $8\pi$ cm. Circle B has an area of $36\pi$ cm$^2$. Find how many times greater the radius of circle B is compared to circle A. Step

1. **Problem 1:** For each pair of points, find the length of the line segment connecting them. Recall the distance formula between two points $ (x_1, y_1) $ and $ (x_2, y_2) $:

1. The problem asks to find the volume of a pyramid with height $h=14$ and base edges $6$ (square base). 2. The formula for the volume $V$ of a pyramid with a square base is

1. Statement of the problem: MNOP is a parallelogram and two consecutive interior angles are $3x-1$ and $2x-1$. 2. In a parallelogram, consecutive angles are supplementary, so thei

1. **State the problem:** We need to prove that the triangle with vertices A(-2,-2), B(4,-2), and C(4,6) is a right triangle by showing that one angle is 90 degrees. 2. **Find the

1. The problem asks for the coordinates of vertex A on the given graph. 2. From the graph description, vertex A is located at the coordinate point $$(-3, 4)$$.

1. **State the problem:** We need to prove that points $A(1,2)$, $B(3,4)$, and $C(0,-1)$ form a scalene triangle. 2. **Calculate the lengths of each side using the distance formula

1. **State the problem:** Find the area of the compound L-shaped figure composed of two rectangles joined together. 2. **Identify individual shapes:** The figure consists of:

1. **State the problem:** We need to find the area of a compound shape formed like an L. The bottom horizontal rectangle measures 9 cm by 3 cm, and the vertical rectangle on the ri

1. **Problem Statement:** Find the area of the L-shaped polygon composed of two rectangles. The top rectangle has dimensions 5 m by 5 m. 2. The bottom rectangle has dimensions 7 m

1. **State the problem:** Find the area of the L-shaped polygon composed of two rectangles with given side lengths. 2. **Analyze the shape:** The shape consists of two rectangles:

1. **State the problem:** We need to find the area of the given L-shaped compound shape consisting of two rectangles. 2. **Identify dimensions:**

1. **State the problem:** We need to prove that the triangle with vertices A(2,3), B(8,11), and C(0,17) is isosceles. 2. **Recall the property of an isosceles triangle:** It has at

1. **Problem 1: Find angle $x$ in a regular pentagon where $x$ is the angle at vertex $B$ between sides $AB$ and diagonal $BD$.** - A regular pentagon has 5 equal interior angles a

1. The problem is to express the relationship between the lengths of segments $AB$, $BC$, and $AC$ such that $|AB| = |BC| \neq |AC|$. 2. The absolute value notation $|XY|$ represen

1. **State the problem:** We need to prove that the triangle formed by points A(-1,3), B(-4,7), and C(0,4) is isosceles. 2. **Recall:** A triangle is isosceles if at least two of i

1. The problem states a rectangle's area is 15 cm². 2. We know the area $A$ of a rectangle is calculated by multiplying its height $h$ and width $w$: