📐 geometry

1. **Problem statement:** We need to find the size of angle $x$ in a triangle with sides 15.3 mm, 13.86 mm, and 6.48 mm, where there is a right angle (90°) between the sides 6.48 m

1. **Stating the problem:** Given a triangle with points P, L, M, and a point N inside the triangle such that $PN = NM$, we want to analyze the relationship and possibly find lengt

1. The problem asks to find the circumference of a circle with a diameter of 146 cm. 2. The formula for the circumference $C$ of a circle is:

1. The problem asks for the circumference of a circle with a diameter of 17 cm. 2. The formula for the circumference $C$ of a circle is:

1. The problem is to find the diameter of a circle given its length as 21 cm. 2. The diameter of a circle is the length of a straight line passing through the center and touching t

1. **State the problem:** We need to find the circumference of a circle with radius $r = 14.2$ meters. 2. **Formula:** The circumference $C$ of a circle is given by the formula:

1. **State the problem:** We need to find the measure of the unknown angle $x$ given three angles expressed in terms of $x$: $x + 15^\circ$, $3x - 22^\circ$, and $2x - 10^\circ$. 2

1. **Problem statement:** We have a circle with center O and points D, E, F on the circumference. Given angles are \(\angle EDF = x^\circ\) and \(\angle OEF = 4x^\circ\).

1. Problem: Find the image of point A(-10,5) under the translation vector T = (2,6). 2. Formula: Translation of a point $(x,y)$ by vector $T = (a,b)$ results in a new point $(x',y'

1. **Problem statement:** Find the area of the segment AYB of a circle with radius $21$ cm and central angle $\angle AOB = 120^\circ$. 2. **Formula and explanation:** The area of a

1. **Problem Statement:** Given a square ABCD with perpendicular segments PQ and RS intersecting inside it, and the condition DQ = CR, find the area of the square ABCD. 2. **Unders

1. **Problem Statement:** Given a square ABCD with segments PQ and RS inside it, where PQ and RS are perpendicular, and DQ = CR. We need to find the area of the square ABCD. 2. **U

1. **Problem statement:** We have two parallel lines cut by a transversal, creating angles labeled $x^\circ$, $y^\circ$, $z^\circ$, $110^\circ$, and $50^\circ$. We need to find the

1. **Problem Statement:** We have a triangle with points A, B, and C. Points A and B are opposite sides of the Grand Canyon. Point C is 200 yards from A. Angle at B is 87 degrees a

1. **State the problem:** We have two similar cylinders. The smaller one has surface area 95 mm² and volume 60 mm³. The larger one has surface area 245 mm² and we need to find its

1. **State the problem:** We need to find angle $A$ in a triangle given side $b$. 2. **Identify what is missing:** To find an angle in a triangle, we need more information than jus

1. **Problem statement:** Given a triangle with an angle of $23^\circ$ and two sides measuring 22.3 cm (opposite the $23^\circ$ angle) and 16.5 cm, find the length of the third sid

1. **State the problem:** We need to find the smallest angle in triangle PQR with sides 9 cm, 10 cm, and 14 cm. 2. **Identify the smallest angle:** The smallest angle is opposite t

1. **Problem statement:** In triangle ABC, angle B is 90 degrees, and T is the midpoint of BC. Prove that $$AC^2 = AD^2 + 3CD^2$$ using the Pythagorean theorem. 2. **Setup and nota

1. **Problem Statement:** In triangle ABC, angle B is 90 degrees and D is the midpoint of BC. Prove that $$AC^2 = AD^2 + 3 \cdot CD^2$$. 2. **Given:**

1. **Problem Statement:** Complete the missing angles and sides of the oblique triangles given partial data. 2. **Key Formula:** The sum of angles in any triangle is always $$180^\