📐 geometry

1. **State the problem:** Calculate the volume of a cone with radius $r=7$ mm and slant height $l=13$ mm. 2. **Recall the formula for the volume of a cone:**

1. **Stating the problem:** Calculate the area of the shaded triangle in the figure labeled 3. 2. **Understanding the problem:** The problem involves a triangle with some given dim

1. **Stating the problem:** We have two similar triangles, ABC and ADE, with given side lengths: $AB=15$ cm, $BD=7.5$ cm, $BC=5$ cm, and $CF=4$ cm. We need to find the area of tria

1. **Stating the problem:** Calculate the unknown height $x$ in triangle 3 and the unknown side $x$ in triangle 4 using the area formulas.

1. **Problem statement:** Determine the amplitude of the angle $DGF$ where the plane $DGF$ is given by the equation $$5x + 5y + 2z = 10.$$ 2. **Step 1: Identify the vectors forming

1. **State the problem:** We need to find the area of a shaded parallelogram inside a rectangle. The rectangle has height 18 cm and width 9 cm. The parallelogram's base is along th

1. **Problem:** If $\triangle PQR \cong \triangle STV$, which sides and angles have equal measurements? 2. **Solution:**

1. Stating the problem: We have two adjacent angles \(\angle A\) and \(\angle B\) such that \(\angle B > \angle A\) and their ratio is \(1:14\). We need to find the measures of \(\

1. The problem states that angle $\angle A$ is less than angle $\angle B$, i.e., $\angle A < \angle B$. 2. To compare angles, we need the measures of $\angle A$ and $\angle B$.

1. The problem states that two adjacent angles, \(\angle A\) and \(\angle B\), have a ratio of 1:5 and that \(\angle B > \angle A\). We need to find the measures of these angles. 2

1. **Problem statement:** We have a right triangle ABC right angled at A with AB = 4 and AC = 6. E is the orthogonal projection of A on BC. S is a direct plane similitude mapping B

1. **State the problem:** We have a trapezium with a left vertical side of 20 cm, a bottom base of 13 cm, and a top-left angle of 129°. We need to find the perimeter of the trapezi

1. **Problem statement:** Given a right triangle ABC with right angle at B, side BC = 16 cm, and angles 58° and 41° at points D and A respectively, find the length of AD. 2. **Unde

1. The problem is to determine how to arrange 10 small boxes inside one larger box. 2. This is a spatial arrangement problem, often solved by considering dimensions and packing.

1. **State the problem:** We have two similar triangles CDE and FGH. We know sides CD = 14, CE = 23, and side HG = 57. We need to find the length of side FG. 2. **Recall the proper

1. **State the problem:** We need to find the radius $r$ of a hemisphere given its volume $V = 78100$ cm$^3$. 2. **Formula for volume of a hemisphere:** The volume of a full sphere

1. **State the problem:** We have two cylinders, A and B. Cylinder A has radius $3.5$ cm and height $7.4$ cm and is $40\%$ full of liquid. The liquid is poured into cylinder B, whi

1. **State the problem:** We are given a midpoint of a line segment and one endpoint, and we need to find the other endpoint. 2. **Formula used:** The midpoint $M$ of a segment wit

1. The problem is to identify and understand the coordinates of points plotted on a coordinate plane. 2. Each point is given as an ordered pair $(x, y)$ where $x$ is the horizontal

1. The problem asks to determine the ordered pairs for each point in the table using the given grid points. 2. The ordered pair for a point is written as $(x, y)$ where $x$ is the

1. **State the problem:** We have a rectangle SUVW with area 40 sq cm and a parallelogram RTWX adjacent to it. We need to find the area of the parallelogram RTWX. 2. **Given data:*