📐 geometry

1. **Problem statement:** Find the area of the corridor outside the square classroom ABCD, express it in terms of $x$, then find $x$ and the area of the classroom given the corrido

1. Problem: Calculate the area of a triangle with base $b=10$ units and height $h=6$ units. 2. Formula: The area $A$ of a triangle is given by $$A=\frac{1}{2} \times b \times h$$

1. **State the problem:** We need to find the image of triangle \(\triangle XYZ\) with vertices \(X(8,4)\), \(Y(4,6)\), and \(Z(8,14)\) after two transformations: reflection across

1. The problem asks for the size of the reflex angle given a smaller angle of approximately 45 degrees. 2. Recall that a reflex angle is the larger angle formed by two rays, greate

1. **State the problem:** We have a circle given by the equation $$x^2 + y^2 + x - 15 = 0$$ and one end of its diameter is at the point $(2, -3)$. We need to find the coordinates o

1. **State the problem:** We need to find the equation of a circle that passes through the points $(0,0)$ and $(4,2)$, and whose center lies on the line $x + y = 1$. 2. **Formula f

1. **Problem Statement:** Find the equation of a circle that touches the coordinate axes at points $(a,0)$ and $(0,a)$. 2. **Understanding the problem:** A circle touching the coor

1. Bài toán yêu cầu so sánh các góc của tam giác ABC dựa trên độ dài các cạnh cho trước. 2. Quy tắc quan trọng: Trong tam giác, góc đối diện với cạnh dài hơn sẽ lớn hơn.

1. **State the problem:** Find the equation of a circle that touches the coordinate axes at points (1,0) and (0,1). 2. **Understand the problem:** A circle touching the coordinate

1. **State the problem:** Find the equation of a circle that touches the x-axis at the point $(3,0)$ and passes through the point $(1,2)$. 2. **Understand the problem:** A circle t

1. The problem is to understand how to determine that the third angle in a triangle is 30 degrees. 2. The key rule is that the sum of the interior angles in any triangle is always

1. **Stating the problem:** We have a circle with center O and points P, Q, R, S on the circumference forming quadrilateral PQRS. Outside the circle, triangle PST is connected with

1. **State the problem:** We need to find the value of $x$ and several angle measures given that $AR \cong RO \cong SA$ and the angles $m\angle AMR = 3x + 20$ and $m\angle ONR = x

1. **State the problem:** We are given a circle with center S and points R, A, O, M on the circumference. We know AR \cong RO \cong SA, and the angle measures m\angle AMR = 3x + 20

1. **State the problem:** We have four rays from the origin forming angles. One angle is 24° between the vertical right ray and an upward ray. We need to find the missing angles $x

1. **State the problem:** We need to find the translation that maps Figure Q onto Figure R. 2. **Identify coordinates:** Approximate coordinates of Figure Q vertices are (6,6), (8,

1. **State the problem:** We need to find the perimeter of a sector with a central angle of $70^\circ$ and radius $8$ cm. 2. **Formula for perimeter of a sector:** The perimeter $P

1. **State the problem:** We need to find the combined area of the dark pieces of wood on a cutting board. 2. **Given data:**

1. **Problem:** Given the scale 1.5 inches = 2 feet, find the actual lengths of each room from their drawing lengths. 2. **Formula:** Use the scale factor to convert drawing length

1. The problem asks: Which angle is vertical to $\angle 2$? 2. Vertical angles are pairs of opposite angles made by two intersecting lines. They are equal in measure.

1. **State the problem:** We have a triangle divided into two smaller triangles by a line from one vertex to the opposite side. The left smaller triangle has a top side length of $