📐 geometry

1. Problem 1: Find the bearing from F to G on the square grid map. 2. To find the bearing, first determine the horizontal and vertical distances between points F and G using the sc

1. **Problem 1: Find cos \(\theta\) and hypotenuse \(d\) in the right triangle.** Given base segments 6.2 cm and 3.8 cm, total base = \(6.2 + 3.8 = 10\) cm.

1. **Problem 1:** The ratio of three angles in a triangle is 1:2:3. Find the measure of each angle in degrees. 2. Let the three angles be $x$, $2x$, and $3x$.

1. **Problem 1: Area of quadrilateral EFGH** The quadrilateral has sides EH = 10 cm (vertical), HG = 11 cm (horizontal), and GF = 5 cm (right vertical segment tilted). Right angles

1. The problem asks to find the triangle congruent to \(\triangle ABC\) with vertices in the correct order. 2. Given triangles:

1. **State the problem:** We are given a triangle ABC with angles at B and C measuring 70° and 30° respectively. We need to find the size of angle AC, which is the angle at vertex

1. **Problem 1: Prove every segment has a unique midpoint.** A midpoint $M$ of segment $AB$ is a point on $AB$ such that $AM = MB$.

1. The problem asks which congruence rule explains why the two triangles sharing side UW are congruent. 2. The triangles share side UW, so side UW is congruent to itself by the Ref

1. **State the problem:** We need to determine which congruence rule explains why triangles UVT and VWT are congruent. 2. **Analyze the given information:**

1. **State the problem:** We need to find the total area of a shape composed of two semicircles side-by-side, each with a diameter of 2 cm. 2. **Identify the radius:** The radius $

1. The problem asks to calculate the area of the shape formed by two shaded quarter circles, each with radius 2 cm. 2. Each quarter circle has an area of \( \frac{1}{4} \pi r^2 \).

1. The problem asks to find the area of a semicircle with a diameter of 21 inches. 2. Recall the formula for the area of a full circle: $$A = \pi r^2$$ where $r$ is the radius.

1. The problem asks to calculate the area of a shape formed by two quarter circles, each with radius 2 cm, combined to form a figure-eight or hourglass shape. 2. Each quarter circl

1. **Problem statement:** A 10 cm by 10 cm by 10 cm cube has a 5 cm by 5 cm square hole cut right through it. Find the volume of the remaining solid.

1. نبدأ بتوضيح خاصية التوازي الخاصة طالمــا: إذا كان لدينا مثلث وقسمنا ضلعين بنسبة معينة، فإن المستقيم المار بنقطتين على هذين الضلعين يكون موازيًا للضلع الثالث إذا تحقق تناسب معين

1. **Problem statement:** We have four towns A, B, C, and D with given positions relative to A:

1. **Problem statement:** We consider a rectangle OABE in an oriented plane with OA = 2 and angle between vectors \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) equal to \(\fr

1. **Problem Statement:** Given quadrilateral ABCD with AC and BD intersecting at E, AB \perp BC, CF \perp BD, and \angle BCF = \angle ACD, prove: (a) \triangle ABC \sim \triangle

1. **Problem statement:** In triangle PQR, given that PQ = PR, MN is parallel to QR, M and N lie on PQ and PR respectively, M is the midpoint of PQ, MQ = RS, and MZS and NRS are st

1. The problem states that the angles of a triangle are in the ratio 5 : 6 : 7. 2. Let the common ratio factor be $x$. Then the angles can be expressed as $5x$, $6x$, and $7x$.

1. Problem: Calculate the length of a tangent drawn to a circle of radius 8 cm from a point 10 cm from its center. Step 1: Let the circle have center O and radius $r=8$ cm.