📐 geometry

1. **Stating the problem:** We have a triangle with two marked angles: one is 81° and the other is $k$. The triangle has two pairs of equal sides, indicating it is an isosceles tri

1. **Problem Statement:** You have several points each with an attached vector of length $r$, and you want to find the surface defined by the ends of these vectors. 2. **Understand

1. **State the problem:** We need to find the values of angles $a$, $b$, and $c$ given the angles $49^\circ$ and $83^\circ$ in a figure with intersecting lines. 2. **Use the proper

1. The problem is to understand how to work with points or functions on different grids. 2. A grid is a coordinate system where points are plotted using coordinates $(x,y)$.

1. **Problem Statement:** Find the area of the curved surface of a cylinder with radius $r=14$ cm and height $h=36$ cm, and express the area in square meters as well.

1. **Problem Statement:** Find the largest area of the region bounded by the circle given by the equation $$x^2 + y^2 = a^2$$ and the vertical line $$y = c$$ where $$-a < c < 0$$.

1. The problem is to accurately draw angles with given directions such as 20°, S 70° W, 160°, S 20° W, N 70° W, 340°, 110°, 270°, 180°, and N 70° E. 2. To draw these angles, we use

1. **Stating the problem:** (a) Identify two congruent shapes from the given polygons.

1. **Problem:** Determine which quadrilateral always has equal diagonals. 2. **Understanding the problem:** We are given four types of quadrilaterals: parallelogram, rhombus, recta

1. The problem involves finding the length of a rectangle placed on a number line with its left side at 1 and right side at 3. 2. To find the length of the rectangle, we use the fo

1. **State the problem:** We need to determine which triangles are similar to triangle $\triangle ABC$ with sides $AB=0.9$, $BC=1$, and $AC=1.5$. Similar triangles have proportiona

1. **State the problem:** We need to determine which triangles are similar to triangle ABC with sides 5, 7, and 8. 2. **Recall the similarity rule for triangles:** Two triangles ar

1. **Stating the problem:** We need to find the length of the diagonal of a rectangle with length 8 and width 6. 2. **Formula used:** The diagonal $d$ of a rectangle can be found u

1. **Problem statement:** We have a quadrilateral field ABCD with given sides AB = 85 m, AD = 72 m, BD = 129 m, and angles \(\angle BDC = 39^\circ\) and \(\angle BCD = 60^\circ\).

1. **Problem Statement:** We have a square ABCD with points Q, S, R, and P on its sides. Two line segments QS and RP intersect at point O inside the square. Given lengths are: $QO=

1. **Problem Statement:** We have triangle $\triangle TUV$ with circumcenter $D$. The perpendicular bisectors $AD$, $BD$, and $CD$ meet at $D$. Given lengths are $CD=30$, $BV=74$,

1. **Problem Statement:** We have triangle $\triangle TUV$ with circumcenter $D$, where $\overline{AD}$, $\overline{BD}$, and $\overline{CD}$ are perpendicular bisectors of the sid

1. **Problem Statement:** Find the values of angles $x$ and $y$ given the angles $44^\circ$, $52^\circ$, $y^\circ$, $x^\circ$, and $97^\circ$ in a geometric figure. 2. **Key Concep

1. The problem describes a rectangle ABCD with sides AB parallel to DC and BC parallel to AD. 2. By definition, a rectangle is a quadrilateral with four right angles and opposite s

1. **State the problem:** We need to find the measure of angle E in a pentagon with angles given as follows: $\angle A = 5x$, $\angle B = 152^\circ$, $\angle C = 3x - 10$, $\angle

1. **State the problem:** We need to find the area of a composite figure consisting of a rectangle on top of a semicircle. 2. **Identify the dimensions:**