📐 geometry

1. **State the problem:** We need to find the value of angle $x$ in a triangle inscribed in a circle, where one angle is given as $22^\circ$ and the triangle is connected to the ce

1. The problem asks to find the value of $x$ in a circle with two intersecting chords forming angles $x+10$ and $146^\circ$. 2. When two chords intersect inside a circle, the oppos

1. **State the problem:** Given that D is the midpoint of AC, AE ≅ CF, AB ⟂ DE, and BC ⟂ DF, prove that ∠A ≅ ∠C. 2. **Given:**

1. **State the problem:** Given that \(\overline{AB} \cong \overline{BC}\) and \(\angle ABE \cong \angle CBD\), prove that \(\triangle BED\) is an isosceles triangle. 2. **Recall t

1. **Problem Statement:** Given parallelogram KLMJ with diagonals intersecting, the segments of the diagonals are given as: - Segment J to intersection: $3y - 5$

1. **State the problem:** Given a parallelogram WXYZ with sides labeled as follows:

1. **Problem statement:** We need to find the cost of cementing the inside walls and base of a cylindrical water tank with an internal radius of 2.5 m and depth 4 m, at a rate of 6

1. **Problem statement:** Given points $A(1, -3, 2)$, $B(2, 5, 3)$, and $C(4, -3, 5)$ in space $Oxyz$, verify if $I(3, 1, -4)$ is the midpoint of segment $BC$. 2. **Formula for mid

1. **Problem statement:** Given isosceles triangle $\triangle ABC$ with $AB = AC$. $M$ is the midpoint of $BC$. Prove: a) $\triangle ABM$ is the angle bisector of $\angle BAC$ and

1. **Problem statement:** We have a rectangular yard ABCD with sides AB = 5m and AD = 12m.

1. **Problem statement:** Given isosceles triangle $\triangle ABC$ with $AB = AC$. $M$ is the midpoint of $BC$. Prove: a) $AM$ is the angle bisector of $\angle BAC$ and $AM \perp B

1. **Problem statement:** Find the sum of the interior angles of a pentagon. 2. **Formula:** The sum of interior angles of a polygon with $n$ sides is given by:

1. **Đề bài:** Cho tam giác ABC vuông tại A. Gọi M là trung điểm của cạnh BC. Trên tia đối của tia AM lấy điểm E sao cho ME = MA. Chứng minh: a) Tam giác AMB = tam giác EMC

1. **Nêu bài toán:** Cho tam giác ABC vuông tại A. Gọi M là trung điểm của cạnh BC. Trên tia đối của tia AM lấy điểm E sao cho ME = MA. Chứng minh: a) Tam giác AMB = tam giác EMC

1. **State the problem:** Find the farthest distance from the point $(12, 2)$ to the circle given by the equation $$x^2 + y^2 + 6x - 16y + 24 = 0.$$\n\n2. **Rewrite the circle equa

1. **Problem 1: Identify the hypotenuse in a spherical triangle with angles** $A=90^\circ$, $B=100^\circ$, $C=92^\circ$. 2. In spherical triangles, the side opposite the largest an

1. **State the problem:** Determine if a spherical triangle can be constructed with sides $a=170^\circ$, $b=100^\circ$, and $c=80^\circ$. 2. **Recall the spherical triangle inequal

1. **Problem Statement:** Given a spherical triangle with angles $A=130^\circ$, $B=110^\circ$, and $C=140^\circ$ on a sphere of radius $R=15$ m, determine the spherical excess.

1. **Stating the problem:** We have a truncated regular quadrilateral pyramid (a frustum) with height $h=6$ cm.

1. **Problem statement:** Given a circle $(O)$ with radius $R=6$ cm and a point $E$ outside the circle such that $OE=9$ cm. Two tangents $EA$ and $EB$ touch the circle at points $A

1. **Problem statement:** Given a circle $(O)$ with radius $R=6$ cm and a point $E$ outside the circle such that $OE=9$ cm. Two tangents $EA$ and $EB$ touch the circle at points $A