📐 geometry

1. **State the problem:** We need to find the volume of a rectangular prism made up of unit cubes. The prism has dimensions 3 cubes wide, 4 cubes tall, and 3 cubes deep, with each

1. **Problem statement:** In triangles \(\triangle JLK\) and \(\triangle DEF\), given \(JK = 5\), \(m\angle JKL = 60^\circ\), \(DE = 2x - 3y\), \(m\angle DEF = x + y\), and \(KL =

1. **Problem 18:** Classify the polygon with vertices $S(7, 2)$, $T(1, 2)$, $U(3, -3)$, and $V(5, -3)$ and find its area. 2. **Step 1: Identify the polygon type.**

1. **State the problem:** We have two intersecting lines at point E, creating angles (5x)° and 65° that are opposite each other. 2. **Identify the relationship:** Vertical angles f

1. **Problem Statement:** We have points $A=(0,1)$, $B=(0,0)$, and $C=(1,0)$ in $\mathbb{R}^2$. An isometry $f$ maps these points to new locations in three scenarios. We want to de

1. **State the problem:** Find the value of $x$ in a triangle with angles $60^\circ$, $50^\circ$, and $x$. 2. **Formula:** The sum of angles in any triangle is always $180^\circ$.

1. **State the problem:** We have triangle ABC with AB = AC = 15 cm, CP perpendicular to AB, and CP = 9 cm. We need to find the shaded area formed by the arc centered at B passing

1. **Problem Statement:** (i) Show that the combination of reflections in two parallel lines, distance $\frac{d}{2}$ apart, results in a translation through distance $d$ perpendicu

1. **Problem Statement:** Find the intersection points of the circles given by the equations $$x^2 + y^2 = 1$$ and $$(x-1)^2 + (y-2)^2 = 4$$. 2. **Step 1: Write down the equations

1. The problem asks to identify which transformation of triangle $\triangle ABC$ is a rotation of $30^\circ$ about point $P$. 2. A rotation transformation involves turning a figure

1. **State the problem:** We need to find the weight of 5 washers. Each washer is a cylindrical ring with an inner diameter of $\frac{1}{4}$ inch, outer diameter of $\frac{3}{4}$ i

1. **State the problem:** We are given a circle with a chord and three angles: an outside angle labeled $74^\circ$, an inside angle adjacent to the chord labeled $81^\circ$, and an

1. **State the problem:** We need to find the image of triangle \(\triangle STU\) after a dilation centered at the origin with a scale factor of \(\frac{1}{4}\). 2. **Recall the di

1. The problem asks for the sum of the interior angles of a polygon with seven sides, called a heptagon. 2. The formula to find the sum of interior angles of any polygon is:

1. **Problem Statement:** We need to find how many triangles can be formed by drawing diagonals from a common vertex in a regular octagon (8-sided polygon). 2. **Formula and Explan

1. **Problem:** Find the missing side lengths of a right triangle with a 45° angle, hypotenuse 6, opposite side $v$, adjacent side $u$. 2. **Formula:** In a 45°-45°-90° triangle, t

1. **Problem:** Find $x$ in a right triangle with legs 4 and $x$, hypotenuse 10. 2. **Formula:** Use the Pythagorean theorem: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse.

1. **Stating the problem:** We are given a circle with points A, B, C, D, E on the circumference. We know $m\angle ACB = 66^\circ$ and the length $mAB = 132$. We need to find $m\an

1. **Problem statement:** Given a circle with center C, points Q and S on the circle, and points R and T outside the circle. Segments QR and SR are tangent to the circle at Q and S

1. **Stating the problem:** We have a right triangle with hypotenuse labeled $x$, one leg labeled $7\sqrt{33}$, and a perpendicular segment from the right angle vertex to the hypot

1. **State the problem:** Verify the area calculation of the irregular polygon. 2. **Recall previous result:** The total area was calculated as $46$ km$^2$ by summing areas of two