📐 geometry

1. **State the problem:** We are given a point $P(1, -2, 1)$ and a plane defined by the equation $x + 2y - 2z = \alpha$, where $\alpha > 0$. The distance from $P$ to the plane is 5

1. **Problem 1: Construct quadrilateral ABCD where AB \cong CD and AD \cong BC.** Step 1: Draw angle \(\angle DAB\) using a protractor and straightedge.

Problem: Draw a chord in a circle with radius $5$ that subtends a central angle of $60^{\circ}$.\n\n1. Given.\nGiven a circle with center $O$ and radius $r=5$.\nWe want the chord t

1. **State the problem:** We need to construct a rectangle where one diagonal divides the opposite angles into 50 degrees and 40 degrees. 2. **Recall properties of a rectangle:** A

1. The problem is to understand how to draw a chord in a circle. 2. A chord is a straight line segment whose endpoints both lie on the circle.

1. The problem states that PQR is a common tangent to two circles touching externally at point O, and OTU is a straight line. 2. Given angles are \(\angle POS = 50^\circ\) and \(\a

1. **State the problem:** We are given the volume of a cylinder as 402 cubic units and the radius as 4 units. We need to find the height $h$ of the cylinder. 2. **Recall the formul

1. The problem asks for the surface area of a rectangular prism with dimensions 2 units (height), 6 units (width), and 8 units (length). 2. The formula for the surface area $SA$ of

1. **State the problem:** Find the surface area of a square pyramid with base side length 6 m, height 4 m, and slant height 5 m. 2. **Calculate the base area:** The base is a squar

Problem: Find the surface area of the hexagonal prism with side length $s=1.5$ and height $h=6$. 1. The surface area formula for a prism is $\text{SA} = 2A_{base} + P_{base}h$.

1. The problem involves a quarter circle centered at point S inside rectangle PQRS, where PQ = 35 m and QR = 12 + x m. 2. The arc KL is part of the quarter circle, and we want to f

1. **State the problem:** We are given a triangle ABC with sides AB = 4 cm, AC = 5 cm, and BC = 6 cm. We need to show that $\cos A = \frac{1}{8}$. 2. **Recall the Law of Cosines:**

1. Let's start by stating the problem: We want to find the formula for the area of a cylinder. 2. A cylinder has two circular bases and a curved surface connecting them.

1. **Problem Statement:** Calculate the area of oblique triangles using Heron's Formula given the lengths of sides $a$, $b$, and $c$. 2. **Heron's Formula:** The area $A$ of a tria

1. Problem: Construct triangle ABC with sides BC = 7 cm, CA = 5 cm, AB = 5 cm. Step 1: Draw segment BC = 7 cm.

1) Problem: Given that \(\triangle ABC \cong \triangle DEF\), answer the following: 1.a) Find the sequence of rigid transformations that take \(\triangle ABC\) to \(\triangle DEF\)

1. The problem asks if lines $j$ and $k$ are parallel and to explain the reasoning. 2. From the graph description, lines $j$ and $k$ are marked as parallel ($j \parallel k$).

1. The problem involves finding the values of angles $h$, $g$, and $f$ formed by two intersecting lines with given angles $100^\circ$ and $33^\circ$. 2. Vertically opposite angles

1. **State the problem:** We have triangle \(\triangle ABC\) with sides \(AB = x\) cm, \(BC = x + 2\) cm, \(AC = 5\) cm, and angle \(\angle ABC = 60^\circ\). We need to find \(x\).

1. **State the problem:** We are given that point P(4,2) is the midpoint of the line segment OPC, where O is the origin (0,0). We need to find the coordinates of point C. 2. **Reca

1. **State the problem:** We need to find the length of side $AB$ in triangle $ABC$ where $AC=6.5$ cm, $BC=8.7$ cm, and the angle $ACB=100^\circ$. 2. **Identify the formula:** Use