📐 geometry

1. **Problem:** Show that segments AB and CD are proportional given points dividing them and solve for unknowns in proportional segments. 2. **Proportional segments rule:** If two

1. **Stating the problem:** We need to find the volume of a cylindrical ring (a hollow cylinder) with outer diameter 30 cm, inner diameter 24 cm, and height 31 cm.

1. **State the problem:** Find the volume and surface area of a composite right prism made of two stacked rectangular prisms.

1. **State the problem:** We have a parallelogram DSOG with sides DO and OG labeled as expressions in terms of $p$: DO = $2p + 9$ and OG = $3p - 6$.

1. **State the problem:** We are given the cosine of angle A in a triangle with vertices A, B, and C, and the formula for cosine of angle A is given by the law of cosines: $$\cos A

1. **State the problem:** Find the circumference and area of a circle with radius $7$ cm, using $\pi = \frac{22}{7}$.\n\n2. **Formulas:**\n- Circumference $C = 2\pi r$\n- Area $A =

1. The problem asks: What is an angle whose vertex is on the circle and whose sides are chords of the circle? 2. Let's analyze the options:

1. The problem asks for the measure of angle $XYZ$ in the given figure. 2. The figure shows a circle with center $O$, points $X$ and $Z$ on the circle along a horizontal line throu

1. The problem is to plot the points A'(-2,3), B(1,-1), and C(3,-4) on a coordinate plane and understand the position of the number -8 in the bottom-right corner. 2. To plot points

1. **State the problem:** We have two circular arcs with the same center O and an angle of 120° each. The smaller arc has radius 1 cm, and the larger arc has radius 5 cm. We need t

1. **State the problem:** We have a sector AOB of a circle with an arc length of $8\pi$ cm. When folded, the sector forms a cone. We need to find the radius of the base of this con

1. **Problem Statement:** We have a cylinder inscribed in a sphere of radius $R=3$. We want to find the height $h$ of the cylinder that maximizes its volume.

1. **Problem statement:** We have two gardens sharing a side AB: one is a rectangle (apples) and the other a regular hexagon (roses). The total wire length around both gardens exce

1. **Problem:** Athul walked along the outer edge of a rectangular park with length 30 m and width 23 m. Find the total distance he walked, which is the perimeter of the rectangle.

1. **Problem Statement:** We have two rectangles placed such that the diagonal starts from the right top corner of the first rectangle and ends at the left bottom corner of the sec

1. The problem asks: How many triangles are in the given geometric sketch? 2. The sketch is a large rectangle divided into 3 rows and 3 columns, making 9 smaller rectangles.

1. **Problem statement:** We have a sector DAB of a circle with center A and radius 18 cm.

1. **Zadání problému:** Máme tři útvary: modrý čtverec, žlutý kruh a zelený trojúhelník. Úkolem je vyznačit 6 bodů tak, aby:

1. **State the problem:** Given a circle with radius $OR = 18$ cm and central angle $\angle SOR = 155^\circ$, find:

1. **Problem statement:** We have a sector AOB of a circle with center O and radius 16 cm.

1. The problem is to find the area of a triangle given its base and height. 2. The formula for the area of a triangle is: