📐 geometry

1. **Problem statement:** Given the number line with points B at -3, O at 0, I at 1, and A at 4, and the line \(\Delta\) defined by points (0, I).

1. **State the problem:** We need to find the surface area of a cylinder with radius $r=4$ and diameter $d=15$. 2. **Recall the formula:** The surface area $A$ of a cylinder is giv

1. **Problem Statement:** Prove that triangles $\triangle HGI$ and $\triangle CID$ are congruent given: - $\angle HGI \cong \angle CID$

1. **State the problem:** We have an unshaded rectangle with perimeter 10 cm, height 3 units, width 2 units, and a larger shaded rectangle formed by joining two copies of the unsha

1. **State the problem:** We need to find the area of the shaded section of a trapezium with a smaller right triangle inside it. 2. **Identify the shapes and dimensions:**

1. **Problem statement:** Calculate the length $x$ in the given triangle with angles 70°, 20°, and side lengths 7 and 4. 2. **Formula used:** Use the Law of Sines: $$\frac{a}{\sin

1. **State the problem:** We need to find the coordinates of point $X$ on segment $CD$ such that the area of triangle $AXD$ equals the area of triangle $BXC$. 2. **Set up the probl

1. **State the problem:** We are given two triangles ABC and DEC where AB is parallel to DE, and the triangles are similar. We need to find the length of AB. 2. **Recall the proper

1. **State the problem:** We have two similar triangles ABC and DEF. We know the sides of triangle DEF: DE = 9 cm, EF = 6 cm, and the sides of triangle ABC: BC = 3 cm, AB = x cm. W

1. **Problem statement:** Aiden and May each place rocks on their side of a river and measure distances to a tree on the opposite bank. We need to determine if their triangles are

1. **State the problem:** Find the total surface area of a combined solid made by joining a triangular prism on top of a parallelogram prism along a congruent rectangular face.

1. **State the problem:** A knight moves forward 3.6 cm from the center of one square to another, then moves diagonally across to the center of the destination square. We need to f

1. **State the problem:** We have a right triangle with vertices A, B, and a third vertex. The side opposite angle $\alpha$ at vertex B is $d=24$, the hypotenuse (side opposite the

1. **Problem statement:** We have a square ABCF and a parallelogram CDEF on the Cartesian plane. Given points F(10,12), D(26,10), and E(18,0), we need to find the coordinates of po

1. **Stating the problem:** We are given points F(10,12), D(26,10), and E(18,0) as part of a polygonal chain with points A, B, C, D, E, and F connected in a specific sequence formi

1. **Problem Statement:** We are given a right-angled triangle OQP with sides OM = 13 cm, MN = 5.5 cm, ON = 12 cm, NQ = 36 cm, and variables $x$ and $y$ representing lengths OP and

1. **State the problem:** We need to find the angle $\theta$ that the slide makes with the ground. The slide forms a right triangle where the height is 107 cm and the hypotenuse (s

1. **Problem statement:** In the right triangle ABC with right angle at B, sides AB = 7 units and BC = 6 units, identify the acute angle at vertex C. 2. **Recall:** In a right tria

1. The problem asks to find the acute angles in a right triangle with vertices A, B, and C, where B is the right angle. 2. Given: side opposite B (hypotenuse) = 7, side opposite C

1. **Stating the problem:** We need to find the area of a kite-shaped quadrilateral with diagonals intersecting at right angles. The kite has sides 13, 13, 15, and 15, and one diag

1. The problem asks to construct a line segment congruent to the segment $\overline{GH}$.\n\n2. To solve this, we need to understand that two line segments are congruent if they ha