📐 geometry

1. **Problem:** Determine if two triangular signs with sides 20 cm, 30 cm, 40 cm and 40 cm, 60 cm, 80 cm are similar. 2. **Formula and rule:** Triangles are similar if their corres

1. **State the problem:** A projector creates a 30cm wide image when it is 50cm from the screen. We want to find the width of the image when the projector is moved back 20cm, makin

1. **State the problem:** We have a sector-shaped water fountain with radius 10.5 m and central angle $\frac{\pi}{2}$ radians. We need to find:

1. **State the problem:** We need to find the total surface area of a composite solid made by joining a cylinder and a cone. 2. **Given data:**

1. **Problem Statement:** Find the surface area of a cylinder and a cone. 2. **Formulas:**

1. **State the problem:** We have a corridor with an actual length of 18 meters, and it is drawn to a scale of 1 : 150. We need to find the length of the corridor in the drawing. 2

1. **Stating the problem:** We have a right triangle with legs of lengths 12 cm and 44 cm, and the hypotenuse labeled as $5y$ cm. We need to find the value of $a$ (assuming $a$ is

1. **Stating the problem:** Given trapezium KLMN with angles \(\angle K = 120^\circ\) and \(\angle M = 110^\circ\), and \(NL = NM\). Point R lies on the extension of \(NL\) beyond

1. **Problem statement:** Given three vertices of a parallelogram $R(-2,-1)$, $S(2,1)$, and $T(0,-3)$, find the coordinates of the fourth vertex. 2. **Formula and rule:** In a para

1. **State the problem:** Given a circle with radius $7$ cm and diameter $TC$, find the circumference, area, various angle measures, arc measures, arc lengths, and the area of the

1. **State the problem:** We have a right triangle $\triangle PQR$ with right angle at $R$. A segment $QS$ is drawn perpendicular to $PR$, intersecting at $S$. We need to complete

1. **State the problem:** We have a right triangle $\triangle PQR$ with a right angle at $R$. Point $S$ lies on segment $PQ$ such that two smaller right triangles are formed inside

1. **Problem Statement:** We are given a right triangle \(\triangle DFG\) with \(FE\) perpendicular to the hypotenuse \(DG\). We need to determine which triangles are similar to \(

1. **State the problem:** We have a right triangle with base $a$ and hypotenuse 18. A segment from the right angle perpendicular to the hypotenuse divides it into two parts, the la

1. **Problem 1: Which triangles are similar to \(\triangle DEF\)?**\n\nGiven the description, \(\triangle DEF\) is a right triangle inside a larger right triangle \(\triangle DFG\)

1. **Problem Statement:** Given triangle $\triangle PQR$ with point $S$ on side $PQ$ such that $PS$ is perpendicular to $SR$ and $RS$ is perpendicular to $QR$, complete the table o

1. **State the problem:** We have a right triangle with base $a$ and hypotenuse 18. A segment from the right angle perpendicular to the hypotenuse divides it into two parts, the la

1. **State the problem:** Find expressions for the total surface area and volume of a rectangular packing crate with dimensions:

1. **State the problem:** We need to find the difference in volume between two cubes built by Hector and Karin with side lengths 12 in. and 14 in. respectively.

1. **Problem statement:** From an external point $P$, tangents $PA$ and $PB$ are drawn to a circle with center $O$. Given that $\angle PAB = 50^\circ$, find $\angle AOB$. 2. **Key

1. **Problem Statement:** Find the area of a figure composed of a rectangle and one semicircle added on the right side, with another semicircle removed on the left side. The rectan