📐 geometry

1. **Problem Statement:** Find the unknown angle $x$ in each polygon given the other interior angles. 2. **Formula:** The sum of interior angles of a polygon with $n$ sides is give

1. **Problem 1: Area of the ring-shaped shaded part** We have two circles: a large circle with radius $6$ m and a smaller inner circle with radius $4.1$ m. The shaded area is the a

1. **State the problem:** Calculate the circumference of a trundle wheel with diameter 40 cm.

1. The problem is to find the length $x$ of the hypotenuse in each right triangle given the legs. 2. We use the Pythagorean theorem: $$x = \sqrt{a^2 + b^2}$$ where $a$ and $b$ are

1. **Stating the problem:** We need to find the length of the side $CD$ of the quadrilateral $ABCD$ given the coordinates of points $C$ and $D$.

1. **Problem statement:** Given a pyramid $S.ABCD$ with base $ABCD$ a trapezoid where $AD = 2BC$. Points $E, F, Q$ are midpoints of edges $SA, SD, AB$ respectively. (a) Prove that

1. **Find the equation of the line of intersection of the planes:** Given planes:

1. **State the problem:** We need to find the size of angle $k$ in two triangles that share a common side, with given angle $81^\circ$ and pairs of equal sides. 2. **Identify the k

1. The problem is to find the area of a triangle. 2. The formula for the area of a triangle is $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$.

1. **Stating the problem:** We have a large square divided into four equal smaller squares arranged in a 2x2 grid. The bottom-right smaller square is further divided into three par

1. The problem is to find the value of $x$ given the lengths 6 cm and 16 cm, but the exact relationship or figure is not specified. 2. Since the problem is unclear, let's assume it

1. **Problem statement:** Given points A, B, C, D on a circle with center O, lines AOBE and DCE are straight, and CO = CE. Given \(\angle AOD = 69^\circ\), find \(x = \angle CEO\).

1. The problem is to understand and apply Lamis Theorem. 2. Lamis Theorem is a result in geometry related to triangles and circles, often used to find lengths or prove properties i

1. **Problem Statement:** Given $AB=11$, $\angle ABD=10^\circ$, $\angle ADB=20^\circ$, and $\angle ACD=60^\circ$, find the value of $BC^2 + AD^2 - AC^2$. 2. **Understanding the pro

1. **State the problem:** We have a right-angled triangle ABC with the right angle at B. The hypotenuse AC is 15 cm, and angle BAC is 38°. We need to find the length of side BC. 2.

1. ปัญหาคือหาค่า $x$ และ $y$ เมื่อมุมที่ส่วนโค้งรองรับคือ $(2x)^\circ$ และ $(5y)^\circ$ และมุมที่จุดศูนย์กลางคือ $80^\circ$\n\n2. กฎสำคัญคือ มุมที่จุดศูนย์กลาง (central angle) มีขน

1. ปัญหาคือหาค่า $x$ เมื่อมุมที่ส่วนโค้งรองรับคือ $(2x)^\circ$ และมุมที่จุดศูนย์กลางคือ $80^\circ$ 2. กฎสำคัญคือ มุมที่จุดศูนย์กลางจะมีขนาดเป็นสองเท่าของมุมที่ส่วนโค้งรองรับ ดังนั้

1. **State the problem:** We need to write the equations of three circles A, B, and C based on their centers and sizes from the graph description. 2. **Recall the formula for a cir

1. The problem is to understand the Midpoint Theorem and its converse. 2. The Midpoint Theorem states: If a line segment joins the midpoints of two sides of a triangle, then this s

1. **Problem:** The area of a trapezium is 34 cm², one parallel side is 10 cm, and the height is 4 cm. Find the length of the other parallel side. 2. **Formula:** The area $A$ of a

1. The user requested graphs for the previous question on bearings. 2. Since no specific function or equation was provided for graphing, and bearings typically involve angles and d