📐 geometry

1. **State the problem:** We have a right triangle DEF with \(\angle E = 90^\circ\), \(DE = 45\), and \(\angle F = 60^\circ\). We need to find the length of segment \(DF\). 2. **Id

1. **Problem statement:** Solve for the angle $x$ in a triangle with sides 2, 3, and 9, where $x$ is the angle opposite the side of length 9. 2. **Formula used:** We use the Law of

1. **Problem statement:** We have a right triangle with one leg measuring 7 cm and the hypotenuse measuring 25 cm. We need to find: a) The length of the other leg.

1. **State the problem:** Find the area of the given irregular polygon with right angles. 2. **Understand the shape:** The figure looks like a large rectangle with a smaller rectan

1. **State the problem:** We have a kite QRST with angles \(\angle T = x^\circ\), \(\angle S = 40^\circ\), \(\angle R = 3x^\circ\), and \(\angle P = 120^\circ\). Line PQU is straig

1. The problem states that the volume of a 3D rectangular solid formed by combining two cubes is 1458 cm³, and we need to find the unknown side length $x$ cm. 2. The volume $V$ of

1. Problem: The user asks to sketch a graph in the plane, but no specific function or equation is provided. 2. Since no function or equation is given, we cannot sketch a graph or p

1. **Problem Statement:** We have a rectangular prism with vertices D, A, B, C, E, F, G, H, and a point J on edge EF dividing it in ratio 2:1.

1. Énoncé du problème : On a un triangle ABC avec AB = 9 cm, AC = 5 cm, BC = 6 cm.

1. **Problem Statement:** Find the value of $x$ given that the area of sector $OAPB$ is $\frac{5}{18}$ of the area of the circle with center $O$.

1. **State the problem:** We have points A(-1, 2) and B(11, 6) on a circle with center P(7, k). M is the midpoint of AB, and line l passes through M and P.

1. **State the problem:** Find the midpoint of the missing term and calculate the equation of each point of vertices given points T(-4,6), P(2,4), A(5,10), E(0,0), and points S, R,

1. Problem: Mary hiked 9 km north and 12 km east. How far was she from her starting point? 2. This is a right triangle problem where the legs are 9 km and 12 km, and we want to fin

1. **State the problem:** We have a rectangular land with perimeter 170 m, length 15 m more than width, divided into three parts P, Q, and R. We need to find the area of the larges

1. **State the problem:** We have a circle $C_1$ with diameter endpoints $A(-2,18)$ and $B(14,6)$. We need to find the equation of $C_1$.

1. **Problem statement:** We have a circle $C_1$ with diameter endpoints $A(-2,18)$ and $B(14,6)$. We need to find the equation of $C_1$. Then, given another circle $C_2$ centered

1. **State the problem:** We need to find the length of segment $LW$ in a right triangle $LYW$ where $LY = 3.5$ units and $AY = 7.0$ units, with $A$ on $LY$ and $AY$ perpendicular

1. **Problem statement:** We have an isosceles triangle $\triangle FOR$ with $FO = 7.0$ units, $FR = 10.0$ units, and $OR = 11.2$ units. Segment $OR$ is parallel to segment $AN$. W

1. **Problem Statement:** We are given a circle with diameter \(\overline{AB}\) and points \(C\) and \(D\) on the circle. The angle between radii \(PA\) and \(PD\) is \((7x+1)^\cir

1. **Problem Statement:** We are given a circle with center $P$ and two diameters $\overline{AD}$ and $\overline{BE}$. We need to find the value of $k$ given the angle $\widehat{CA

1. **Problem Statement:** Find the length of side AB in a right triangle \(\triangle ABC\) where \(\angle B = 90^\circ\), hypotenuse \(AC = 25\) cm, and one leg \(BC = 24\) cm.