📐 geometry

1. **Problem statement:** Calculate the capacity of the chocolate filling inside the solid, which consists of a rectangular prism with a triangular prism on top. 2. **Understanding

1. **State the problem:** Calculate the total surface area of the chocolate painted object, which consists of a closed cylinder with an open hemisphere inside its top.

1. **State the problem:** We are given two similar triangles \(\triangle CGR \sim \triangle SDA\) with side lengths expressed as \(AD = -2x - 8\), \(GR = 6\), \(CR = 12\), and \(AS

1. **State the problem:** We are given two similar triangles \(\triangle AFP \sim \triangle HDG\) with side lengths \(AP = 15\), \(FP = 11\), \(DG = -8x - 7\), and \(GH = -6x + 4\)

1. Problem: Draw a line parallel to a given line so you can see what “parallel” means. 2. Key idea (parallel lines): Parallel lines have the same direction, so their slopes are equ

1. **State the problem:** We have a right triangle with hypotenuse length 5 ft. The longer leg is 1 ft longer than the shorter leg.

1. The problem is to understand and draw a parallel line. 2. Parallel lines are lines in a plane that never meet; they are always the same distance apart.

1. **State the problem:** We have a right triangle with the shorter leg as $(x - 9)$ inches, the longer leg as $x$ inches, and the hypotenuse as $(x + 9)$ inches. We need to find t

1. **State the problem:** We need to find the area of a composite figure made by joining two rectangles. 2. **Identify the dimensions:**

1. **State the problem:** We need to find the area of a composite figure made by joining two rectangles. 2. **Identify the rectangles and their dimensions:**

1. **State the problem:** Find the perimeter and area of a rectangle with vertices at (2,4), (2,6), (8,6), and (8,4) on the coordinate plane. 2. **Formula for perimeter and area of

1. **State the problem:** Find the perimeter and area of the triangle with vertices at points (4,1), (6,1), and (5,8) on the coordinate plane. 2. **Formula for perimeter:** The per

1. **Problem:** Find the value of $x$ given the angles $(x - 5)^\circ$, $(2x - 7)^\circ$, and $21^\circ$ in a triangle. 2. **Formula:** The sum of interior angles in a triangle is

1. State the problem.\n\nA dog is leashed to a corner of a house with leash length $20$ ft, and the running area is a quarter-circle outside the corner. Find the running area.\n\n2

1. **Problem Statement:** (i) Prove that if there are $n+1$ points on one line in a projective plane, then the total number of points is $n^2 + n + 1$.

1. **State the problem:** We are given an angle \(\angle JML\) with measure 85°. 2. The angle is split into two parts: \((6x + 2)^\circ\) and \((4x + 3)^\circ\).

1. **State the problem:** We need to find dimensions of a pyramid with a square base that result in a volume of 36 cubic units. 2. **Formula for the volume of a pyramid:**

1. **State the problem:** We are given a prism with surface area 54 square inches. We want to find the surface area of a similar prism that is smaller by a scale factor of 3. 2. **

1. The problem asks to find the surface area of a composite shape consisting of a cylinder and a rectangular prism. 2. The formula for the surface area of a cylinder is $$SA_{cyl}

1. **Problem:** Find the midpoint of the line segment with endpoints A(-4,5) and B(-5,0). 2. **Formula:** Midpoint $M$ of segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is give

1. **Problem:** Reflect square BCDE with vertices B(-6,7), C(-2,6), D(-3,2), E(-7,3) across the y-axis. 2. **Formula and rule:** Reflection across the y-axis changes a point $(x,y)