📐 geometry

1. **State the problem:** We need to find three integer dimensions $x$, $y$, and $z$ of a cuboid such that its surface area is 340 cm². 2. **Formula for surface area of a cuboid:**

1. **State the problem:** We need to reflect the polygon ABCD with vertices A(2, -2), B(4, -2), C(4, -4), and D(2, -4) across the line $y = x$. 2. **Formula for reflection across $

1. **Problem statement:** Calculate the lateral area of a regular octagonal prism where the perimeter of the base is 24 cm and the height is one-fourth of the perimeter. 2. **Formu

1. **State the problem:** We have a right angle composed of two adjacent angles labeled as $(8x)^\circ$ and $(6x + 6)^\circ$. We need to write an equation to find $x$ and then solv

1. **State the problem:** Find the area of the shaded region, which is the area of the sector of a circle with radius $32$ and central angle $45^\circ$ minus the area of the triang

1. **State the problem:** We have a right triangle ABC with a right angle at D on segment AC. Given lengths are $AC=20$ and $CD=12$. We need to find the length $BC$. 2. **Understan

1. **Problem statement:** We have a right triangle PQR with a right angle at point S on the base PR. Given lengths are $PQ=5$ and $PR=7$. We need to find the length $PS$. 2. **Unde

1. **State the problem:** We have a right triangle EFG with a right angle at F. A perpendicular segment FH is dropped from F to the base EG. Given EH = 14 and GH = 6, we need to fi

1. **Problem statement:** Find the total length of metal bar needed to make the part of the swing formed by points A(0,5), B(-4,0), and C(4,0). Then find the slopes of AB and AC, a

1. **Problem statement:** A square is inscribed in a circle with area $\pi$ square units. Find the area of the square. 2. **Given:** Area of circle $= \pi$.

1. **Problem statement:** We have a large cone with radius $r=10$ cm and slant height $l=12$ cm.

1. The problem asks to determine if given sets of angles or side lengths form a unique triangle, more than one triangle, or no triangle. 2. Important rules for triangles:

1. **Problem Statement:** Determine which given area calculations for the shapes with specified side lengths are incorrect. 2. **Approach:** For each shape, calculate the area base

1. **Problem:** Determine whether each pair of angles is True (T) or False (F) based on their relationship in the figure with two parallel lines and a transversal. 2. **Key angle r

1. **Problem Statement:** Reflect the given rectangle across the x-axis. 2. **Understanding Reflection Across the x-axis:**

1. **Problem Statement:** Determine whether the given angle name statements are True (T) or False (F) based on the figure with points A-H on a horizontal line and points J-M on a d

1. **Problem Statement:** Determine whether the given pairs of lines in the tetrahedron are intersecting, perpendicular, parallel, or skew. 2. **Recall definitions:**

1. **State the problem:** We have a right triangle ABC with right angle at A. Points D and E lie on AC and BC respectively, with \(\angle CAB = \angle DEC = 90^\circ\). Given lengt

1. **Problem statement:** We have a right triangle with a hypotenuse of length 15, an angle of 51°, and the side opposite this angle labeled as $x$. We need to find the length of s

1. **Problem statement:** We need to find the length of the unknown side $x$ in a right triangle where the other two sides are 2 and 5. 2. **Formula used:** According to the Pythag

1. **State the problem:** We need to determine if a triangle with sides $a$, $b$, and $c$ is a right triangle using the Pythagorean theorem. 2. **Recall the Pythagorean theorem:**