📐 geometry

1. **State the problem:** We have a square with side length 8 units. One vertex E is at (1, 3), and vertex F lies on the same horizontal line (same y-coordinate) as E but 8 units t

1. **Problem Statement:** Given two circles with center O and diameter OB of the smaller circle, and angle ÂBC = 45°, prove: (i) AC is parallel to OD

1. **Problem statement:** Given quadrilateral ABCD with AB = AD, a perpendicular from A to CD meets CD at X, AX and BD intersect at Y, and YC = YD. Prove that triangles CXY and XYD

1. **State the problem:** We have a quadrilateral ABCD with an inscribed circle P tangent to all sides. Given sides AB = 12 units, DC = 9 units, AD = 4 units, and BC = 4 units, we

1. **Problem Statement:** A circular clock has radius $r=20$ cm. A chord forms an equilateral triangle with two radii. We need to find the percentage area of the smaller segment cr

1. Problem statement: The height of a right prism is increased by 50%, while the base area remains the same. We need to find the percentage increase in the volume of the prism. 2.

1. **State the problem:** We need to complete the construction of parallelogram PQRS given the diagonal PR. 2. **Step 1:** Draw the diagonal PR using a ruler.

1. **State the problem:** We need to determine which set of three side lengths can form a right-angled triangle. 2. **Recall the Pythagorean theorem:** For a triangle with sides $a

1. The problem asks to identify the types of lines represented on a door made of seven rectangular pieces of wood joined together with some wood on top forming a zig zag pattern. 2

1. **State the problem:** We need to find the difference in volumes between a cylindrical block and a rectangular block of wood. 2. **Calculate the volume of the cylindrical block:

1. The teacher starts by explaining the concept of volume as the amount of space occupied by a 3D object. 2. Using 45 identical cubes, the teacher constructs a cuboid with a square

1. The teacher begins by introducing the concept of volume as the amount of space occupied by a 3D object. 2. Using 45 identical cubes, the teacher constructs a cuboid with a squar

1. **Problem 1:** Given two triangles ABC and CDE with ACE and BCD straight lines, AB parallel to DE, AC = 8 cm, CD = 15 cm, and DE = 20 cm, find the unknown side length $y = CB$.

1. The problem is to find the volume of a solid, which depends on the shape and dimensions given. 2. Identify the shape of the solid (e.g., cube, cylinder, sphere, cone, prism).

1. The problem asks for the ratio between the areas of two similar polygons given the ratio of their perimeters is 4 : 9. 2. For similar polygons, the ratio of their areas is the s

1. The problem involves a pentagonal prism with a base edge length of 4.5 cm and a prism length (height) of 12 cm. 2. To find the volume of the prism, we need the area of the penta

1. **Problem statement:** Find the volume of each prism given the dimensions and formulas. 2. **Part a:** Cube with side length 11 cm.

1. The problem asks to find the circumference of a ball (sphere) given its volume of 5.6 dm³. 2. Recall the formula for the volume of a sphere: $$V = \frac{4}{3} \pi r^3$$ where $r

1. The problem is to find the volume of a shape or object given dimensions 5 and 6. 2. However, volume requires three dimensions (length, width, height) or a formula specifying the

1. The problem is to find the formula for the circumference of a circle. 2. The circumference is the distance around the circle.

1. **State the problem:** We need to find the slant height $l$ of a cone given its surface area $S = 204.2$ and radius $r = 5$. 2. **Recall the formula for the surface area of a co