📐 geometry

1. Problem 1: Given a triangle with sides $a=\sqrt{3}$, $b=\sqrt{2}$, and angle $a=60^\circ$, find angle $B$. 2. Use the Law of Cosines formula: $$b^2 = a^2 + c^2 - 2ac \cos(B)$$ b

1. مسئله: در مثلث ABC، با توجه به اینکه CE=3 است، اندازه ضلع AB را بیابید. 2. اطلاعات داده شده: زاویه \(\angle ABD = 45^\circ\)، \(\angle BAD = 60^\circ\)، و CE=3.

1. مسئله: در یک نمودار دایره‌ای، زاویه مرکزی \(\alpha\) متناظر با ۲۰ درصد از کل افراد جامعه است. باید مقدار زاویه \(\beta\) را پیدا کنیم. 2. فرمول: مجموع زوایای یک دایره برابر است

1. **State the problem:** We need to find the length of side $x = UV$ in a right triangle $\triangle UTV$ where $\angle U = 90^\circ$, $UT = 64$, and $\angle T = 70^\circ$. 2. **Id

1. The user requests problems that are more complex and high-definition (hd), avoiding simple shapes like squares and rectangles. 2. To meet this request, we can consider problems

1. Problem: Find the perimeter and area of a rectangle with length 15 cm and width 8 cm. 2. Formula:

1. **Problem Statement:** Find the area of the shaded region, which is an annulus (ring shape) with outer radius $7$ cm and inner radius $3$ cm. 2. **Formula:** The area of an annu

1. **Problem statement:** We have a circle with radius 5 cm and a chord PQ of length 8 cm. Tangents at points P and Q intersect at point T. We need to find the length of segment TP

1. The problem states: "All have 4 edges." We need to understand what this means in a mathematical or geometric context. 2. In geometry, an edge is a line segment joining two verti

1. **State the problem:** We need to find the total volume of a composite solid made up of a right circular cone on top of a hemisphere. The radius of both the cone's base and the

1. Problem statement: A circle with center O has chord AE, and OD is drawn from O perpendicular to AE with right angle at D. 2. Problem statement: Another chord BF meets AE at E, a

1. Problem statement: We have a cyclic quadrilateral WXYZ (points on a circle) with diagonals WY and XZ intersecting inside, and the given small angles at the vertices are: at X: 7

1. **Stating the problem:** We have a cyclic quadrilateral WXYZ inscribed in a circle with angles at each vertex split into two parts: at W angles $a^\circ$ and $b^\circ$, at X ang

1. **Stating the problem:** We have a cyclic quadrilateral WXYZ inscribed in a circle with given angles: at X, angles 70° and 28°; at Z, angles 31° and 51°; and unknown angles a°,

1. **Stating the problem:** We have a circle with points W, X, Y, Z on its circumference forming four triangles: WXY, WYZ, XYZ, and WZY. Angles at these points are given or labeled

1. **Problem Statement:** Given a circle with center O and points A, B, C on the circumference, angle AOB at the center is $t$. Angles on the circumference are $\angle ABC = 38^\ci

1. **Problem statement:** Given a circle with diameter AB and a point C on the circumference, angle ABC is 38°. We need to find the measure of angle ACB. 2. **Key property:** In a

1. The problem asks to find the volume of a cube with side length $\frac{4}{9}$ miles. 2. The formula for the volume $V$ of a cube with side length $s$ is:

1. **State the problem:** We need to find the volume of a rectangular prism with side lengths $\frac{3}{5}$ m, $\frac{5}{6}$ m, and $\frac{1}{4}$ m. 2. **Formula for volume of a re

1. **State the problem:** We have a rectangular solid spacer with dimensions length $L=8$ ft, height $H=14$ ft, and width $W=32$ ft.

1. **Problem Statement:** Calculate the volume of titanium needed for one triangular prism bead with a hole, then find the total titanium volume for 500 beads, the weight in pounds