📐 geometry

1. **State the problem:** We are given a triangle with three angles labeled as $ (8x - 4)^\circ $, $ (17x - 23)^\circ $, and $ (3x + 17)^\circ $. We need to find the value of $x$.

1. Problem 6a: Find the composition of transformations $T_{(3,-2)} \circ T_{(1,-1)}$. 2. The rule for composition of translations $T_{(a,b)} \circ T_{(c,d)}$ is:

1. Problem: Write a reflection rule that maps each triangle to its image. 2. Reflection rule means finding the line or axis over which the triangle is flipped.

1. **Stating the problem:** Prove or verify the equation $$\frac{AB}{AE} + \frac{AD}{AF} = 1$$ where points and segments are part of a geometric figure involving parallelogram $ABC

1. **State the problem:** We have two hemispheres and a right prism. The total volume of the two hemispheres equals the volume of the prism. We need to find the value of $p$. 2. **

1. সমস্যাটি হলো: একটি গোলকের ব্যাসার্ধ $R$ এবং একটি লম্ববৃত্তাকার চোঙের ব্যাসার্ধ $r$ এবং উচ্চতা $\frac{9}{2}r$ দেওয়া আছে। গোলক এবং চোঙের আয়তন সমান হলে $R:r$ এর অনুপাত নির্ণয় করত

1. **Problem Statement:** Prove that points D, A, C, and E lie on the same circle, i.e., quadrilateral DACE is cyclic. 2. **Key Property:** A quadrilateral is cyclic if and only if

1. **Problem statement:** Find (a) the length CD in terms of r and sin \(\theta\), (b) the perimeter of sector CAB given \(r=4\) and \(\theta=\frac{1}{6}\pi\), and (c) the area of

1. **Problem statement:** Given rectangle ABCD with AB > BC, line xy passes through B and is parallel to diagonal AC. Line xy intersects AD at point E. Prove that quadrilateral ACB

1. The problem asks to plot the points (-2,1), (0,3), (3,0), and (3,5) on the Cartesian coordinate system. 2. Each point is represented as $(x,y)$ where $x$ is the horizontal coord

1. **Problem Statement:** Given a parallelogram ABCD with diagonals intersecting at O, points P and Q are midpoints of AO and BC respectively. Given that $\angle A = \angle DPO$ an

1. مسئله: اندازه پاره‌خط AB در دایره مثلثاتی برابر $\sqrt{3} + \sqrt{2}$ است. نقطه A روی محور طول‌ها (محور x) در سمت منفی قرار دارد و نقطه B روی محیط دایره در ربع اول است. هدف یافت

1. The problem is to find the volume of a prism. 2. The formula for the volume of a prism is $$V = B \times h$$ where $B$ is the area of the base and $h$ is the height of the prism

1. The problem is to understand what a prism is in geometry. 2. A prism is a solid geometric figure with two parallel, congruent bases connected by rectangular faces.

1. The problem is to find the volume of a solid. 2. To find volume, we need to know the shape and its dimensions. Common formulas include:

1. The problem involves finding the area of a trapezoidal cross-section with top width $4.8$ m, bottom width $1.2$ m, and height $1.5$ m. 2. The formula for the area $A$ of a trape

1. **Stating the problem:** We have a trapezoidal prism with the trapezoid base having top side $4.8$ meters, bottom side $1.2$ meters, and height $1.5$ meters. We want to find the

1. The problem asks to solve the 2nd question of the given figure, but since the figure is not provided, I will explain how to approach a typical 2nd question in math problems invo

1. The problem asks to match each polar coordinate point $(r, \theta)$ with one of the labeled points A, B, C, or D on the graph. 2. Recall that in polar coordinates, a negative ra

1. **Problem 8:** \(\angle A\) is obtuse and \(\angle A = (x + 20)^\circ\). Find the limits of \(x\). - An obtuse angle is greater than 90° and less than 180°.

1. **Stating the problem:** We have two parallel lines PQ and RS cut by a transversal XY. We need to identify corresponding angles and find the values of angles \(\angle a\) and \(