📐 geometry

1. **Stating the problem:** We have a quadrilateral with vertices A(-8,-3), B(13,-3), C(8,9), D(-3,9). We need to represent it on a Cartesian plane, identify the polygon, calculate

1. **Problem statement:** In triangle $ABC$, $AD$ is perpendicular to $BC$ and satisfies the relation $AD^2 = BD \times DC$. We need to prove that $\angle BAC = 90^\circ$. 2. **Giv

1. **Stating the problem:** We have a polygon with vertices at points $(1,9)$, $(4,9)$, $(4,7)$, $(2,4)$, and $(2,1)$. We want to increase its size by a factor of $\frac{3}{2}$. Th

1. **State the problem:** We are given a triangle ABC with sides AC = $7\sqrt{3}$ cm, AB = $\sqrt{6}$ cm, and BC = $a$ cm. The angle at vertex A is 45°. We need to show that $a = \

1. **State the problem:** We have a right triangle with a hypotenuse of length 8.5, and the base is divided into two segments of lengths 3.2 and 6.8 by a height $x$ perpendicular t

1. **State the problem:** We need to find the value of $\cos E$ in a right triangle $EFG$ where the right angle is at vertex $F$. The sides adjacent to $F$ are $EF = 18$ and $FG =

1. **State the problem:** We have a rectangular prism ABCDEFGH with the sum of all edge lengths equal to 28 cm and total surface area equal to 13 cm². We need to find the length of

1. **State the problem:** We have a rectangular prism ABCDEFGH with the sum of all edge lengths equal to 28 cm and total surface area equal to 13 cm². We need to find the length of

1. **State the problem:** We have a rectangular prism ABCDEFGH with edges summing to 28 cm and total surface area 13 cm². We need to find the length of the diagonal EC. 2. **Define

1. **State the problem:** We have a rectangular prism ABCDEFGH with the sum of all edge lengths equal to 28 cm and total surface area equal to 13 cm². We need to find the length of

1. **State the problem:** We have a rectangular prism with edges summing to 28 cm and total surface area 13 cm². We want to find the length of the diagonal $EC$. 2. **Define variab

1. **State the problem:** Two students build boxes: one a cube with edge 60 cm, the other a rectangular prism with width $w$ cm, height 30 cm, and depth 45 cm. Both boxes have the

1. The problem asks to identify the quadrant of the point (8,7). 2. Recall the quadrant rules for points $(x,y)$ in the Cartesian plane:

1. **Énoncé du problème :** Montrer que la droite (DC) est l'image de la droite (AB) par la translation de vecteur \(\overrightarrow{BC}\).

1. Problem: To find the length of side BC in two similar triangles ABC and ADE, given |AE|=2, |EC|=3, and |ED|=4. 2. Since triangles ABC and ADE are similar, corresponding sides ar

1. **Stating the problem:** We have two right triangles with given side lengths and an unknown segment $x$. We need to find $x$ in each case.

1. **State the problem:** We are given the circumference $C$ of a circle as 119.32 inches and need to find the diameter $d$ of the circle, rounded to the nearest hundredth. 2. **Fo

1. **Stating the problem:** We have four points on a horizontal line: $A(1,2)$, $M(a,b)$, $N(c,d)$, and $B(10,17)$ arranged from left to right. We want to understand the relationsh

1. **Stating the problem:** We have two right triangles with sides labeled and relationships given. We want to find the values of $x$ and $y$ using the given ratios and the fact th

1. **State the problem:** We need to find the volume of a sphere with a diameter of 16 inches. 2. **Formula for the volume of a sphere:**

1. **Problem statement:** We are given two similar equilateral triangles with areas 396 cm² and 275 cm² respectively.