📐 geometry

1. **State the problem:** We need to find the length of the flight of stairs, which is the hypotenuse of a right triangle with a height of 22 m and a base of 39 m. 2. **Formula use

1. مسئله: در مثلث $ABC$ که $AB=AC$ و زاویه $\hat{A}=80^\circ$ است، نقطه $D$ داخل مثلث قرار دارد به طوری که $\hat{D}\hat{B}C=10^\circ$ و $\hat{D}\hat{C}B=30^\circ$. باید ثابت کنیم م

1. مسئله: سه نقطه A، B و C روی یک دایره قرار دارند. دو ارتفاع AH و BG یکدیگر را در نقطه D قطع می‌کنند و امتداد AH با دایره در نقطه M برخورد می‌کند. باید ثابت کنیم پاره‌خط‌های DH و

1. **Problem Statement:** Find the values of variables $x$ and $y$ in Figure 17, where two intersecting lines form vertical angles labeled $(9x + 12)^\circ$, $3x^\circ$, and $(4y -

1. **State the problem:** We need to find the values of $x$ and $y$ in two figures using angle relationships such as supplementary angles, alternate exterior angles, and correspond

1. **Problem:** Find the coordinates of triangle DEF after each transformation. Given vertices: D(4, -1), E(5, 2), F(1, 2).

1. **State the problem:** We need to find the measure of angle $\angle B$ in a right triangle with vertices $B$, $C$, and $A$. The side $BC$ is perpendicular to $CA$, so $\angle C$

1. **State the problem:** We have an irregular polygon with given side lengths: 16 cm (top horizontal), 6 cm (right vertical), 11 cm (bottom horizontal segment), 4 cm (downward ver

1. **State the problem:** We need to find the ratio of the area of a sector with central angle $\alpha$ radians to the area of the circle inscribed in that sector. 2. **Recall form

1. The problem is to draw a triangle, which is a geometric figure with three sides and three angles. 2. Since this is a drawing request and not a mathematical problem, we cannot pr

1. The problem asks for the radius of a circle given its circumference of 135 cm. 2. The formula for the circumference $C$ of a circle is:

1. State the problem: Given an isometric sketch of a solid, produce its oblique sketch using an oblique projection method. 2. Formula and projection rule:

1. **Problem statement:** Prove that the square ABCD and the rectangle CMEF have equal areas given that CF = CP and \(\angle CBP = \angle BMP\).

1. **Problem statement:** A rectangle is divided into four smaller rectangles. Three of them have areas 6 cm², 18 cm², and 36 cm². We need to find the area of the shaded rectangle.

1. **Problem statement:** We have two chords WF and HG intersecting inside a circle at point E. Given angles are $\angle W = 55^\circ$ and $\angle G = 71^\circ$. We need to find th

1. **Problem statement:** Find the measure of the unknown arc or angle indicated in the circle with given angles 52° and 121°. 2. **Relevant formula:** In a circle, the sum of angl

1. **Problem statement:** Calculate the surface area of a cylinder. 2. **Formula:** The total surface area $A$ of a cylinder is given by

1. **State the problem:** We need to determine the placement of 6 sprinklers to cover a 9 by 16 tile plot, where each sprinkler waters 24 adjacent tiles. 2. **Understand sprinkler

1. **Stating the problem:** We have two triangles \(\triangle PQR\) and \(\triangle VUT\) that are similar (\(\triangle PQR \sim \triangle VUT\)). We know the angles and some side

1. **State the problem:** We need to find the perimeter and area of a rectangle with vertices at (5, -1), (5, -5), (-3, -5), and (-3, -1). 2. **Recall formulas:**

1. **State the problem:** Find the area and perimeter of the rectangle with vertices $(-3, 3)$, $(-3, -7)$, $(4, -7)$, and $(4, 3)$. 2. **Identify the lengths of the sides:** Since