📐 geometry

1. **State the problem:** We are given two inscribed angles in a circle, one at point U labeled $16x + 26^\circ$ and another at point R labeled $42^\circ$. We need to find the valu

1. **State the problem:** We need to find the volume of an L-shaped prism composed of two rectangular blocks. 2. **Identify dimensions:** The smaller block has dimensions 4 cm (wid

1. The problem is to find the perimeter of a shape. 2. The perimeter is the total distance around the shape, calculated by adding the lengths of all its sides.

1. **Stating the problem:** Construct triangle ABC with sides AC = 9 cm, BC = 7 cm, and angle BAC = 90°. 2. **Understanding the problem:** We have a right triangle with angle BAC =

1. **Stating the problem:** Construct triangle ABC where side AC = 9 cm, side BC = 7 cm, and angle BAC = 90°. 2. **Understanding the problem:** We have a right triangle with angle

1. The problem is to find the perimeter of a shape. 2. The perimeter of a polygon is the sum of the lengths of all its sides.

1. **Problem Statement:** Calculate the size of angle $x$ at point $A$ on the circumference of a circle with center $O$, given that angle $D$ on the circumference is $142^\circ$. 2

1. **State the problem:** We are given two vertices of a triangle, $A(2,1)$ and $B(3,-2)$, and the area of the triangle is 5. The third vertex $C$ lies on the line $y = x + 3$. We

1. **Problem:** Two circles with radii 6 cm and 8 cm have centers 10 cm apart. Find the area common to both circles. 2. **Formula and explanation:** The area of intersection of two

1. **Problem:** Two circles with radii 6 cm and 8 cm have centers 10 cm apart. Find the area common to both circles. 2. **Formula:** The area of intersection of two circles with ra

1. **State the problem:** We are given two vertices of a triangle, $A(2,1)$ and $B(3,-2)$, and the area of the triangle is 5. The third vertex $C$ lies on the line $y = x + 3$. We

1. **Problem Statement:** We have two similar trapezoids. The larger trapezoid has a vertical side of length 8 and a bottom base of length 12. The smaller trapezoid has a vertical

1. **Problem Statement:** We have a square cardboard of side 80 cm. We cut out squares of side 10 cm from each corner to form an open box by folding the sides up. We want to find t

1. **State the problem:** We have a right triangle KLM with a right angle at L. Side KL = 47, side LM = 94, and we need to find the angle $x$ at vertex M. 2. **Identify the sides r

1. **Stating the problem:** We have a right triangle KLM with a right angle at L. The sides KL and LM are given as 47 and 94 respectively, and we need to find the angle $x^\circ$ a

1. **Stating the problem:** We need to prove that the segment (CA) is the perpendicular bisector of the segment [DB] in two different ways, and optionally find a third way.

1. **Problem:** Explain why opposite angles in a parallelogram are always congruent. 2. **Definition and properties:** A parallelogram is a quadrilateral with both pairs of opposit

1. **State the problem:** We need to find the volume of a prism with length 36 mm and a triangular cross-section with base 18 mm and height 29 mm. 2. **Formula for volume of a pris

1. **Problem statement:** We have a shape made from a cylinder and a hemisphere. The radius $r$ of both is 8 cm. The total volume is given as $\frac{2684}{3} \pi$ cm$^3$. We need t

1. **State the problem:** We have a triangle with three angles: $2x + 16$, $x + 1$, and $80$ degrees. We need to find the value of $x$. 2. **Recall the triangle angle sum rule:** T

1. **Problem:** Find the value of $x$ in a triangle with angles $(2x+16)^\circ$, $80^\circ$, and $(x+1)^\circ$. 2. **Formula:** The sum of angles in any triangle is always $180^\ci