📐 geometry

1. Problem: The area of a circle is numerically equal to four times its circumference. Find the radius of the circle. 2. Formulae:

1. **Problem statement:** Calculate the volume of a cylinder (walec) with radius $r=5$ cm and height $h=7$ cm. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** We are given two similar triangles, $\triangle CDE \sim \triangle CAB$, and need to find the length $EB$. 2. **Identify known lengths:**

1. **State the problem:** Given that $\overline{AB} \parallel \overline{DC}$, prove that $\triangle ABC \cong \triangle CDA$ using the flowchart proof. 2. **Given:** $\overline{AB}

1. **Problem statement:** Given an acute triangle ABC with altitude AH and a circumscribed circle centered at O with diameter AM. 2. **Part a: Calculate angle $\widehat{ACM}$**

1. **Stating the problem:** Given a triangle ABC inscribed in a circle with center O, points H, N, and M lie on or inside the triangle and circle. Lines AH, BN, and CM are drawn fr

1. **State the problem:** We have a sector with a central angle of 60° and an unknown radius $r$. Its area equals the area of a circle with radius 7 cm.

1. **State the problem:** Two arcs of the same circle subtend angles in the ratio 3:5. The difference of their arc lengths is 16\pi cm. We need to find the radius of the circle. 2.

1. **State the problem:** Given that $AB \parallel DC$ in parallelogram $ABCD$, complete the flowchart proof to show that $\triangle ABC \cong \triangle CDA$. 2. **Given:** $AB \pa

1. **State the problem:** Given that segment $\overline{AC}$ bisects segment $\overline{BD}$, complete the flowchart proof showing $\triangle ABE \cong \triangle CDE$. 2. **Given:*

1. **State the problem:** Two arcs of the same circle subtend angles in the ratio 3:5, and the difference of their arc lengths is 16\pi cm. We need to find the radius of the circle

1. **State the problem:** We are given angles and lines in a diagram and need to identify pairs of lines or angles based on their relationships: perpendicular lines, parallel lines

1. **State the problem:** Given that segment $\overline{AC}$ bisects segment $\overline{BD}$, complete the flowchart proof to show that $\triangle ABE \cong \triangle CDE$. 2. **Un

1. **State the problem:** Given that segment $\overline{AC}$ bisects segment $\overline{BD}$, complete the flowchart proof showing the congruence of triangles $\triangle ABE$ and $

1. The problem states that triangles ABC and DEF must be congruent based on the information marked in the diagram. 2. To determine if two triangles are congruent, we use congruence

1. The problem asks which congruence theorems or postulates can be used to prove that triangle ABC is congruent to triangle UVW based on the given diagram. 2. Common triangle congr

1. **Problem statement:** Find the value of $x$ where $IJ = x$, $HG = 8$, and $EF = 12$ in trapezoid $FEGH$ with median $JI$. 2. **Formula:** The median of a trapezoid is the segme

1. **State the problem:** We are given a circle with points G, F, V, and H and angles at vertices V (119°), H (65°), and an external angle near G (68°). We need to find the angle a

1. **Introduction to Coordinate Proofs and Trilateration** Start by explaining that coordinate proofs use algebra and geometry together on the coordinate plane to prove properties

1. Let's create a problem involving quadrilaterals. 2. Problem: Given a quadrilateral with vertices at points $A(1,2)$, $B(4,2)$, $C(4,5)$, and $D(1,5)$, find the perimeter and are

1. **Stating the problem:** We have a diamond-shaped quadrilateral ABCD with sides labeled as follows: