📐 geometry

1. **State the problem:** We have two similar trapezoidal prisms. The smaller prism has side lengths 5 mm, 6 mm, and a top edge of 2 mm with height $x$ mm. The larger prism has sid

1. The problem involves calculating the surface area (SA) and perimeter (P) of a rectangle and then using these values to find the total surface area of a related shape. 2. Given:

1. **Stating the problem:** We have a cube [OABCDEFG] with edge length 3 in the coordinate system Oxyz.

1. **State the problem:** We have two similar parallelograms $WXYZ \sim MNOP$. We need to list pairs of corresponding congruent angles, write ratios of corresponding sides, find th

1. **Problem Statement:** Given pairs of corresponding congruent angles and sides: $AZ = XY$, $WX = NO$, and variables $x, y, z$ representing side lengths.

1. The problem involves identifying relationships between line segments in a triangular prism based on the given vertices and edges. 2. We are asked to analyze pairs of segments: J

1. **State the problem:** We have a square pyramid with base vertices G, H, I, J and apex K. We need to determine which pairs of lines among \(\overrightarrow{HI}\) and \(\overrigh

1. **State the problem:** Prove that triangles DEF and D'E'F' are congruent by finding a sequence of transformations that maps DEF onto D'E'F'. 2. **Identify coordinates:**

1. **State the problem:** We need to find the angle of rotation about the origin that maps triangle $\triangle XYZ$ with vertices $X(-4,2)$, $Y(-6,6)$, $Z(-2,6)$ to triangle $\tria

1. **Problem statement:** We have a right triangle ABC with a right angle at B and a smaller triangle ABD inside it. Given $AC = 19$ cm, and angles $\angle B = 46^\circ$ and $\angl

1. **State the problem:** We need to find the length of segment $PQ$ in a right triangle where sides $PS = x + 2$ and $PR = 2x - 1$. 2. **Identify the relationship:** Since $PQ$ is

1. The problem asks to identify the type of center represented by point M in a triangle. 2. The options are Circumcenter, Incenter, Centroid, and Orthocenter.

1. **State the problem:** We are given a triangle XYZ with segment AB as the midsegment. AB is labeled as $3x - 1$ and side XZ is labeled as 34. We need to find the value of $x$. 2

1. **Problem Statement:** Identify which angle is complementary to \(\angle 2\) given the angles formed by two intersecting lines. 2. **Recall definitions:**

1. **State the problem:** We need to find the length of the diagonal $AG$ in a cuboid where edges $AB$ and $AD$ meet at a 75° angle, with $AB=23$ cm and $AD=18$ cm. 2. **Understand

1. **State the problem:** Find the angle $\theta$ in triangle A-5 where the legs are 5 and unknown $\theta$, and the hypotenuse is 13.

1. **State the problem:** We need to find the scale factor that takes rectangle Y to rectangle Z. 2. **Identify the dimensions:**

1. The problem is to explain the concept of length between two points $p$ and $q$ in a coordinate system. 2. The length (or distance) between two points $p=(x_1,y_1)$ and $q=(x_2,y

1. **Problem Statement:** Calculate the lengths $p$ and $q$, the perimeter, and the area of an L-shaped plot of land with given segment lengths. Then find the maximum number of ban

1. **Problem statement:** Calculate the area of a square when its diagonal length is given as $1.6 + a$ cm. 2. **Formula used:** The area $A$ of a square with diagonal $d$ is given

1. **Stating the problem:** We need to find the area of the sector ALB of a circle with center L, where AL = 14 cm, angle LAC = 25°, and AC = BC.