📐 geometry

1. **Problem statement:** Given a right triangle ABC with a right angle at B, M is the midpoint of AC. From M, MH is drawn perpendicular to AB at H, and MK is drawn perpendicular t

1. The problem involves finding the value of $x$ given angles in a circle, where angles such as $x^\circ$, $2x^\circ$, $3x^\circ$, $-x^\circ$, and $130^\circ$ appear in various con

1. **Problem Statement:** Explain the properties of a quadrilateral, specifically focusing on the sum of interior angles and the diagonals forming triangles.

1. **Stating the problem:** We need to find the length of segment $JN$ by using the similarity of two right triangles $\triangle JKL$ and $\triangle JNM$. 2. **Identifying similar

1. **Problem Statement:** We need to find the length of segment $JN$ in the given right triangle configuration. 2. **Given:**

1. **State the problem:** We need to classify triangle $\triangle WXY$ with vertices $W(-7,1)$, $X(0,-4)$, and $Y(2,3)$ as equilateral, isosceles, scalene, or none of the above. 2.

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: