📐 geometry

1. **Problem statement:** Given that triangles \(\triangle CDE\) and \(\triangle FCE\) are similar (\(\triangle CDE \sim \triangle FCE\)), find the value of \(x\) where \(CE = x\).

1. **Problem:** Find the scale factor of Figure B to Figure A given sides of triangles. Given sides:

1. **Problem Statement:** We need to find the image of triangle $\triangle KLM$ after reflecting it over the line $x = -1$. 2. **Given:**

1. **State the problem:** We have a translation transformation $T$ that maps point $P(-2, 2)$ to $P'(0, 0)$. We need to find the image of the point $(5, -1)$ under the same transla

1. Use the Pythagorean theorem to find the unknown sides. **Problem:**

1. **Problem Statement:** Find the length $d$ of the base of the large right triangle given the height is 12 m, one angle is 35°, and there is a smaller triangle inside with a 10°

1. **Problem Statement:** Given a cube with edges of length 6 cm, find:

1. **State the problem:** Find the area of the irregular polygon with given side lengths in miles. 2. **Approach:** We can divide the irregular polygon into simpler rectangles and

1. The problem is to find the area of an irregular polygon composed of a triangle and a stepped rectangular shape. 2. The triangle has a base of 8 km and height of 9 km. The area o

1. **Stating the problem:** We are given a triangle ABC with point D on BC such that \(\overrightarrow{BD}\) bisects \(\angle ABC\) which measures 40°. 2. The angle bisector divide

1. **Problem Statement:** We are given a quadrilateral with points A, B, C, D, E, F such that AD = DB and AF = FC. The lengths DF = 14 and DE = 10 are known. We need to find the le

1. **Problem Statement:** We have triangle ABC with points D, E, and F on its sides. Given that AE = EC and BF = FC, points E and F are midpoints of sides AC and BC respectively. W

1. **Problem Statement:** Given triangle ABC with points D, E, and F on the sides such that AD = DB, AE = EC, EF = 6, and $m\angle AED = 32^\circ$. We need to find $m\angle ECF$. 2

1. **Problem Statement:** We are given triangle $ABC$ with $AE = EC$, $m\angle BAC = m\angle DFE$, $AC = 24$, and $m\angle DFE = 126^\circ$. We need to find the length of segment $

1. **State the problem:** We need to find the perimeter of the shaded ring sector KLNM formed between two circular arcs with central angle 113° and radii 24 cm (outer) and 15 cm (i

1. **Problem statement:** We have two concentric circular sectors OKL and OMN with the same central angle of 113°. The smaller sector radius is 15 cm and the larger sector radius i

1. **Problem Statement:** We have a parallelogram with one side length of 70 mm and the adjacent side length labeled as $t$. The perimeter of the parallelogram is given as 149.2 mm

1. **State the problem:** We need to find the coordinates of point C given the following conditions: - The x-coordinate of C is the same as the x-coordinate of A.

1. **State the problem:** We need to find the size of angle $a$ in a right triangle $PQR$ where \angle $P = 90^\circ$, \angle $R = 30^\circ$, and \angle $Q = 20^\circ$. Angle $a$ i

1. **Stating the problem:** We have a triangle with points A, B, C and a segment DE parallel to side AB. Given lengths are AB = 12, CB = 18, and DE is parallel to AB. We want to fi

1. **Problem Statement:** We need to find the length of segment $LK$ in a diamond-shaped figure with points $L$, $K$, $N$, and $M$. Given that $KL = x$ and $KN = 2x - 3$, and there