📐 geometry

1. **State the problem:** Given that AECF is a parallelogram, triangles ABC and ADC are congruent, and segments EB and FD are congruent, prove that AECF is a rhombus. 2. **Recall d

1. **State the problem:** We are given a right triangle with sides $a$, $b$, and hypotenuse $c$. The Pythagorean theorem states that $$a^2 + b^2 = c^2.$$ We know $c = 5$ and $a^2 =

1. **State the problem:** We need to find the volume of a cone with height $h=19$ yards and radius $r=17$ yards. 2. **Formula for the volume of a cone:**

1. **Stating the problem:** We have a triangle ABC with right angles at B and D. Segment BD is perpendicular to AC, and AB is perpendicular to BD. Angle BDA is a right angle, and a

1. **Problem Statement:** Find the measures of angles $\angle CBD$, $\angle ABD$, and $\angle A$ in triangle $ABC$ with given conditions: $AD \perp BC$ at $D$, $BD \perp AC$ at $D$

1. **State the problem:** We have triangle KLM with angles \(\angle L = 65^\circ\), \(\angle K = 0^\circ\) (which is unusual), and an exterior angle \(\angle LMS = 115^\circ\). We

1. **Problem Statement:** We have an isosceles triangle KLM with sides KL = KM. Points L, M, and S are collinear with M between L and S. The exterior angle at vertex M, adjacent to

1. **Problem Statement:** We are given a rectangle ABCD with angles \(\angle a\) and \(\angle b\) at vertex D. The angle between the horizontal segment DC and diagonal DA is given

1. **State the problem:** We are given a triangle with sides $a=14$, $c=8$, and angle $B=64^\circ$. We need to find side $b$ using the Law of Cosines. 2. **Formula:** The Law of Co

1. **Problem Statement:** We have three scale drawings of a prism: front view (6 units wide by 6 units tall), side view (8 units wide by 3 units tall), and plan view (8 units wide

1. **Problem Statement:** We are given a circle with center $O$ and radius 2. Points $A, B, C, D, E$ lie on or inside the circle. Line segments $AC$ and $BD$ intersect at $E$. Give

1. **Problem Statement:** We have two problems involving triangles and the Pythagoras theorem.

1. **Problem statement:** We have a rectangular oil tank with dimensions 5 m (length), 3 m (height), and 2 m (width). The oil fills the tank to a depth of 1.2 m. We need to find th

1. **State the problem:** We need to find how many smaller cuboid boxes (10 cm by 3 cm by 4 cm) fit completely inside the larger cuboid box (50 cm by 30 cm by 20 cm). 2. **Formula

1. **Problem statement:** We have a pentagon and a triangle sharing a side. We know the pentagon's angles: 107°, 31°, 102°, and 62°, and the triangle's angles: 79° and $k$. We need

1. **Stating the problem:** We have a shape $R$ translated by the vector $$\begin{pmatrix} 2 \\ -6 \end{pmatrix}$$ to get shape $R'$. Then $R'$ is translated to get shape $R''$. We

1. **Problem Statement:** Given a parallelogram ABCD with diagonals intersecting at O, where $\angle CD = 13\frac{6}{3}$ (interpreted as $13 + \frac{6}{3} = 15$ degrees), and $OD$

1. Énoncé du problème : On considère un triangle ABC rectangle en A avec AB = 2 et BC = x.

1. **Problem Statement:** We are given that line X contains the hypotenuses of two similar triangles LMN and PQR. The slope of line X between points (-7, 6) and (-4, 4) is given as

1. **Problem Statement:** Given quadrilateral ABCD with AB \parallel DC, \(\angle BDC = 30^\circ\), and \(\angle BAD = 80^\circ\), find the values of \(x = \angle ABC\), \(y = \ang

1. **Problem statement:** We have a right triangle XYZ with a right angle at vertex Y. The side XY is 9 cm, the hypotenuse XZ is 17 cm, and we need to find the length of side YZ. 2