📐 geometry

1. The problem asks for the first step in constructing a line perpendicular to vector $\overrightarrow{AB}$ passing through an external point $M$ using only a compass and straighte

1. **Problem Statement:** We are given a circle with points F, G, H, and A on its circumference. Triangle FGH is inscribed in the circle. The segment GH measures $21x - 2$ and segm

1. **Problem Statement:** We have a right triangle inscribed in a circle. The triangle has sides labeled 10.4, 8.4, and $x$, with a right angle between the sides 8.4 and $x$. We ne

1. **State the problem:** We have a triangle with angles 68°, 125°, and an unknown angle $x$. We need to find the measure of angle $x$. 2. **Formula used:** The sum of the interior

1. **Problem Statement:** We are given a triangle with angles and sides, and we need to find the measure of angle $x$. 2. **Given Information:**

1. **State the problem:** We are given a triangle with three angles labeled as $3x - 5$, $5x + 5$, and $3x - 15$. We need to find the measure of angle $x$. 2. **Recall the triangle

1. **Problem Statement:** Given $m\angle 2 = 55^\circ$ and $m\angle 3 = 80^\circ$, find $m\angle 4$.\n\n2. **Understanding the figure and relationships:** The angles are part of a

1. **Problem Statement:** Find the area of the shaded sector of a circle with radius $60$ units and arc length $150$ units.

1. **Problem Statement:** Find the area of a segment of a circle with radius $r=40$ and central angle $\theta=165^\circ$. 2. **Formula:** The area of a segment is given by the diff

1. **Problem Statement:** Find the area of the shaded sector of a circle with radius $45$ units and arc length $50$ units. 2. **Formula for arc length:** The arc length $s$ of a se

1. **Stating the problem:** We have a square ABCD with vertices A(3,7), B(3,3), and D(7,7). We need to find the coordinates of the fourth vertex C. 2. **Formula and rules:** In a s

1. **Problem statement:** We have a straight line JKLM with points K, L, M, and angles x and y. Given \( \cos x = \frac{1}{\sqrt{5}} \) and the ratio \( LM : KL = 1 : 2 \), we need

1. **State the problem:** We are given a cube with volume $v = 729$ m$^3$ and side length $x = 9$ m. We need to calculate the surface area of the cube's net in cm$^2$ and express t

1. **Problem Statement:** Find the equation of a circle centered in the first quadrant that is tangent to the lines $x=8$, $y=3$, and $x=14$. 2. **Understanding the problem:** The

1. **State the problem:** Find the center and radius of a circle given the ends of a diameter at points $(18, -13)$ and $(4, -3)$. 2. **Formula used:**

1. **Problem statement:** Find the equation of a circle given its center and radius. 2. **Formula:** The standard form of a circle's equation with center at $(h,k)$ and radius $r$

1. **Problem statement:** We have a right triangle with legs $YZ = 20$ cm and $ZX = 21$ cm, and we want to find the hypotenuse $YX$. 2. **Formula used:** In a right triangle, the P

1. **Problem Statement:** We have a trapezoid with the top base $11$ cm, the bottom base $26$ cm, and one non-parallel side (left side) $17$ cm. We need to find the height $h$ of t

1. **State the problem:** We need to find the volume of air in a balloon with a diameter of 24 cm. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. **State the problem:** Prove that a parallelogram with congruent diagonals has a right angle. 2. **Given:**

1. **Problem:** Determine if the two triangles with one right angle and two equal sides are congruent. 2. **Rule:** Triangles with a right angle and two equal sides (hypotenuse and