📐 geometry

1. **State the problem:** We have a right triangle ABC with a right angle at A. The hypotenuse BC is 36 cm, and the segment from A to the point where the inscribed circle touches t

1. **Stating the problem:** We are given two parallel lines O-S-P and Q-T-R intersected by a transversal M-S-T-N. At point S, the angle is labeled as $(2x + 55)^\circ$, and at poin

1. **State the problem:** We are given two parallel lines cut by a transversal, creating angles \( (6x + 12)^\circ \) and \( 132^\circ \). We need to solve for \( x \). 2. **Identi

1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are \((52 - 2x)^\circ\) and \((22x + 88)^\circ\). We need to s

1. **State the problem:** We need to find the measure of angle $\angle MSP$ given two expressions for angles formed by a transversal intersecting two parallel lines. 2. **Given:**

1. **State the problem:** We need to find the measure of angle $\angle MSP$ given that $\angle MSP = (7x + 60)^\circ$ and $\angle QTR = (108 - 3x)^\circ$. Lines $OQ$ and $PR$ are p

1. **Problem Statement:** A solid metal cylinder of height 10 cm and diameter 14 cm is melted and recast into two cones with volumes in the ratio 3:4, both cones having height 10 c

1. **Problem statement:** We are given a quadrilateral TUVW with angles at T and V known: $\angle T = 102^\circ$ and $\angle V = 59^\circ$. Sides WT and UV are equal.

1. **Problem statement:** We are given a parallelogram KGHJ with angle $\angle K = 38^\circ$. We need to find the size of angle $\angle KGH$. 2. **Recall properties of parallelogra

1. **Problem Statement:** Find the values of $x$ and $y$ that make each quadrilateral a parallelogram based on the given diagonal segment lengths. 2. **Important Property:** In a p

1. **State the problem:** We are given the vertices of a triangle at points $(2,-4)$, $(-5,3)$, and $(-7,0)$. We need to plot these points and draw the triangle formed by connectin

1. **Problem Statement:** We have a right-angled triangle ABC with D and E as midpoints of AB and BC respectively. We want to determine which statements are sufficient to find the

1. **Problem statement:** We have a triangle with vertices at approximately (2,3), (5,3), and (5,7). The triangle is reflected across a vertical line, and two vertices of the refle

1. **Problem statement:** We have two triangles, A and B, on a coordinate grid. Triangle A is translated to form triangle B. We need to find the values of $d$, $e$, and $f$ which a

1. **State the problem:** We start at point A (0,0). Travel 7 km on a bearing of 135° to point B. Then from B, travel 12 km on a bearing of 250° to point C. We need to find the dis

1. **State the problem:** We are given a kite ABCD with sides labeled as expressions involving variables $v$ and $w$: - $AB = 10v - 13$

1. **State the problem:** We need to find the area of a trapezium with bases $a=8$ cm, $b=9$ cm, and height $h=6$ cm. 2. **Formula:** The area $A$ of a trapezium is given by the fo

1. **Problem statement:** Find the values of the angles marked with letters in the given diagrams. 2. **Focus on part (b):** Given $b=80^\circ$ and $a=\frac{3}{2}c$, find angles $a

1. **Problem statement:** We need to find the length of side $AB$ in the triangular prism. Given are angle $\angle F = 24^\circ$, side $FE = 6$ cm, and slant height $EC = 13$ cm. 2

1. **Problem Statement:** Given triangle $ABC$ with midpoints $D$, $E$, and $F$ of sides $BC$, $CA$, and $AB$ respectively, prove that the medial triangle $DEF$ is similar to $ABC$

1. Let's consider the example of a rotation around the origin by an angle $\theta = \frac{\pi}{4}$ (45 degrees) counter-clockwise. 2. The rotation transformation is given by the fo