📐 geometry

1. The problem involves finding the area of shapes with dotted lines, which usually represent heights or altitudes. 2. The key formula for area of a parallelogram or triangle is $\

1. **State the problem:** We are given a circle with a tangent line touching it at one point. The angle between the tangent line and the radius to the point of tangency is $x^\circ

1. **State the problem:** We need to find the measure of angle $m\angle BCD$ in the given circle with points $E, A, B, C, D$ arranged clockwise. 2. **Given information:**

1. **State the problem:** We have a right triangle with angles 30° and 60°, the side opposite the 30° angle is given as $7\sqrt{3}$ cm, and we need to find the hypotenuse $w$. 2. *

1. **State the problem:** We have a right triangle with angles 30°, 60°, and 90°. The side opposite the 30° angle is 12 meters, and we need to find the length of side $n$, which is

1. **State the problem:** We have two triangles, \(\triangle ABC\) and \(\triangle EDC\), sharing vertex \(C\). Given sides are \(AB=9\), \(AC=12\), \(BC=10\), \(ED=k\), \(DC=x\),

1. **State the problem:** Find the approximate area, in square inches, of the shaded region in a rectangle 12 in. wide and 6 in. high containing two side-by-side circles each with

1. **Problem Statement:** Given the coordinates of four locations on a grid where each unit represents 0.75 miles, find the distances between these locations in miles.

1. **State the problem:** Determine if the two triangles described in the first graph (top-left) are congruent and explain why. 2. **Description:** Two triangles share a base with

1. The problem asks where to rotate a shape and where to place it. 2. To rotate a shape, you need a point called the center of rotation.

1. **Problem:** Rotate the triangle with vertices at (2,2), (3,5), (4,2) 90° counterclockwise about the origin. 2. **Formula for 90° counterclockwise rotation:**

1. **Problem Statement:** Find the distance between points A and B on a coordinate grid where point B is 2 units right and 2 units up from point A. 2. **Formula Used:** The distanc

1. **State the problem:** We need to find the area of an irregular polygon composed of rectangles and triangles with given side lengths in millimeters. 2. **Analyze the figure:** T

1. **State the problem:** Find the area of the irregular polygon with given side lengths. 2. **Analyze the figure:** The polygon has vertical and horizontal segments with one diago

1. **Stating the problem:** We are given multiple angle relationships involving expressions with $x$ and constants, mostly pairs of angles that are either opposite (vertical) angle

1. **State the problem:** We need to describe a sequence of rigid motions followed by a dilation with center at (0,0) that maps quadrilateral RSTU to VXYZ. 2. **Given points:**

1. The problem asks to determine if each statement about coordinates and axes is true or false. 2. Statement 1: "In the ordered pair (3, 5), the number 5 is the y-coordinate."

1. The problem asks for the ordered pair that describes the location of point J on the coordinate grid. 2. An ordered pair is written as $(x, y)$ where $x$ is the horizontal coordi

1. **State the problem:** We need to find the area of a semicircle with diameter $d = 24$ kilometers. 2. **Recall the formula:** The area of a full circle is given by $$A = \pi r^2

1. **Problem statement:** Find the area of a semicircle and a quarter circle each with radius $r=13$ miles. 2. **Formulas:**

1. **State the problem:** Find the total area of the given figure with dimensions 3 m, 4 m, and 9 m. 2. **Analyze the figure:** The figure can be divided into a rectangle and a tri