📐 geometry

1. The problem is to find the perimeter of a shape given its dimensions. 2. The perimeter of a polygon is the sum of the lengths of all its sides.

1. **State the problem:** We need to find the length $d$ in the smaller triangle, given angles and a side length in the larger triangle. 2. **Identify known values:** In the larger

1. **State the problem:** We need to find the surface area of a hexagonal prism where each hexagonal face has an area of 23 cm² and the height of the prism is 9 cm. 2. **Recall the

1. **State the problem:** We have a prism with a regular decagon as its cross-section. Each side of the decagon is 2 cm, and the height of the prism is 12 cm.

1. The problem is to identify the visible components of a unicycle based on the description. 2. The unicycle consists of several parts: a large dark purple circular wheel outline,

1. **State the problem:** Given that \(\overline{TV}\) bisects \(\angle SYU\) and \(\angle TSY \cong \angle TUV\), prove that \(\triangle TSY \cong \triangle TUV\). 2. **Identify g

1. **State the problem:** We have three right triangles and need to find the unknown side $x$ in each using the Pythagorean theorem. 2. **Recall the Pythagorean theorem:** For a ri

1. **State the problem:** We need to find the coordinates of point $R(4,5)$ after reflecting it once across the $x$-axis and once across the $y$-axis. 2. **Reflection in the $x$-ax

1. **Problem Statement:** Find the scale factor of the sides for the similar triangles \(\triangle KLI \sim \triangle TUV\).

1. **State the problem:** We are given two parallel lines cut by a transversal, creating segments of lengths 14 and 8 on one side, and 12 and an unknown length $x$ on the other sid

1. **Problem statement:** (i) Starting from a right triangle with sides 1, 1, and $\sqrt{2}$, construct a segment of length $\sqrt{3}$ using straightedge and compass.

1. **Problem Statement:** We have a sequence of figures made by joining polygons with sides of unit length. The first three figures have perimeters 5, 8, and 11 respectively. We ne

1. The problem involves finding the area of a parallelogram with given side lengths and height. 2. The formula for the area of a parallelogram is:

1. **Problem Statement:** (i) Starting from a right triangle with sides 1, 1, and hypotenuse $\sqrt{2}$, construct $\sqrt{3}$ using straightedge and compass.

1. **Problem statement:** Given triangle ABC with points P on AB and Q on AC such that PQ is parallel to BC, prove that $$\frac{|AP|}{|AB|} = \frac{|AQ|}{|AC|} = \frac{|PQ|}{|BC|}.

1. **Problem statement:** (i) Show that if $a,b,c > 0$ and $a^2 + b^2 = c^2$, then $a + b > c$ so that $a,b,c$ can form a triangle.

1. **Problem statement:** (i) Show that if $a, b, c > 0$ and $a^2 + b^2 = c^2$, then $a + b > c$ so that $a, b, c$ can form a triangle.

1. **Problem statement:** We have a circle with center O and a tangent line EDC touching the circle at point D. We need to find the size of angle $z$ formed between the tangent lin

1. **State the problem:** We have a circle with center $O$, a tangent line $ABC$ touching the circle at $B$, and chords $AD$, $BE$, and $CE$. We need to find the size of angle $\th

1. **Problem Statement:** We need to find a formula for the area of trapezoid ABCD in terms of $a$, $b$, and $h$. The trapezoid has one pair of parallel sides: $AB$ and $DC$. Side

1. **State the problem:** We are given a sphere with a radius of 10 meters and asked to find its volume in cubic meters. 2. **Formula for the volume of a sphere:** The volume $V$ o