📐 geometry

1. **Problem statement:** A 10 cm by 10 cm by 10 cm cube has a 5 cm by 5 cm square hole cut right through it. Find the volume of the remaining solid.

1. نبدأ بتوضيح خاصية التوازي الخاصة طالمــا: إذا كان لدينا مثلث وقسمنا ضلعين بنسبة معينة، فإن المستقيم المار بنقطتين على هذين الضلعين يكون موازيًا للضلع الثالث إذا تحقق تناسب معين

1. **Problem statement:** We have four towns A, B, C, and D with given positions relative to A:

1. **Problem statement:** We consider a rectangle OABE in an oriented plane with OA = 2 and angle between vectors \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) equal to \(\fr

1. **Problem Statement:** Given quadrilateral ABCD with AC and BD intersecting at E, AB \perp BC, CF \perp BD, and \angle BCF = \angle ACD, prove: (a) \triangle ABC \sim \triangle

1. **Problem statement:** In triangle PQR, given that PQ = PR, MN is parallel to QR, M and N lie on PQ and PR respectively, M is the midpoint of PQ, MQ = RS, and MZS and NRS are st

1. The problem states that the angles of a triangle are in the ratio 5 : 6 : 7. 2. Let the common ratio factor be $x$. Then the angles can be expressed as $5x$, $6x$, and $7x$.

1. Problem: Calculate the length of a tangent drawn to a circle of radius 8 cm from a point 10 cm from its center. Step 1: Let the circle have center O and radius $r=8$ cm.

1. The problem states that the shaded area is a right triangle inside a square, and its area is 11 m². 2. The diagonal of the square divides it into two congruent right triangles,

1. **Complete the sentences:** 1.1 A parallelogram can be defined as a quadrilateral with both pairs of opposite sides parallel.

1. Problem: Find the height of a cone with radius 6 cm and volume 301.44 cm³. The volume formula for a cone is $$V=\frac{1}{3}\pi r^2 h$$.

1. The problem asks for the new coordinates of the point $(2,3)$ after a transformation or change, but the specific transformation is not given. 2. To find the new coordinates, we

1. State the problem: Prepare a 20-question worksheet on circles for grade 10. 2. Understand the requirements: Questions should carry marks 1, 2, 3, 4, 5.

1. **State the problem:** Given triangle $\triangle PQR$ with vertices $P(1,3)$, $Q(4,1)$, and $R(6,y)$, and $RQ \perp PQ$, solve the following: 2.1 Find the gradient of $PQ$.

1. **Stating the problem:** We have a wooden hanging bookshelf shaped like a house. The top part is a triangle with height 12 cm and side lengths 35 cm. Below it are two rectangula

1. **State the problem:** We have a right-angled triangle with base length $x$ cm and height $(x - 1)$ cm. The area is given as 15 cm$^2$. 2. **Recall the formula for the area of a

1. **State the problem:** We have a figure made from intersecting semicircles each with diameter $\frac{1}{2}$. We need to find the exact area of the shaded portion formed by these

1. **State the problem:** We have two congruent circles with centers $P$ and $Q$. Points $R$, $P$, $Q$, and $S$ lie on a horizontal line such that $RP = QS = 1$. The circles inters

1. Problem: We are given the formula for the area of a parallelogram as $$A = g \cdot h$$, where $A$ is the area, $g$ is the base, and $h$ is the height. 2. We want to express the

1. The problem states that the shape has vertices with values 2 (top-left), 1 (top-right), and 3 (bottom-right), and the equation given is $S_3 = 24$. 2. We interpret $S_3$ as the

1. **Problem statement:** We have a circle with center O and a quadrilateral inscribed in it. We know one angle on the circumference is 25°. We need to find the values of the lette