📐 geometry

1. Problem 9: In triangle $\triangle ABC$, given $AB=\sqrt{2}$ cm, $BC=\sqrt{3}$ cm, and angle $\angle BAC = 60^\circ$. Show that $\angle ACB = 45^\circ$ and find the length $AC$.

1. The problem asks us to find the total surface area of a cylindrical candle with radius $r = 14$ cm and height $h = 16$ cm. 2. Recall that the total surface area $A$ of a cylinde

1. The problem asks to calculate the volume of the given L-shaped prism using the provided dimensions. 2. Interpret the prism structure as two rectangular prisms combined: one larg

1. The problem involves finding the bearing of point A from point B, given a scale drawing where 1 cm represents 100 m. 2. From the description, point A is at the bottom right and

1. The problem asks to find the range of possible lengths for the third side $x$ of a triangle given two sides, using the triangle inequality rule: $$|a-b| < x < a+b$$

1. The problem states that the map scale is 1:4,000,000, meaning 1 cm on the map represents 4,000,000 cm in reality. 2. The distance between two cities on the map is 5 cm.

1. The problem asks if the given sets of side lengths can form a triangle. 2. Recall the triangle inequality theorem: For any triangle, the sum of the lengths of any two sides must

1. **State the problem:** We are given that \(\angle 1 \cong \angle 2\) and we want to prove that lines \(a\) and \(b\) are parallel. 2. **Given:** \(\angle 1 \cong \angle 2\).

1. **State the problem:** Given that \(\angle 1 \cong \angle 2\), prove that lines \(a\) and \(b\) are parallel. 2. **Given:** \(\angle 1 \cong \angle 2\).

1. **State the problem:** We have a circle centered at point $P$ with diameters $\overline{AD}$ and $\overline{BE}$. We need to find the measure of the minor arc $\stackrel{\large{

1. **State the problem:** We have a circle centered at point $P$ with points $A$, $B$, $C$, $D$, and $E$ on the circumference. 2. Given that $\overline{AD}$ and $\overline{BE}$ are

1. **State the problem:** We need to find the measure in degrees of the major arc $\stackrel{\large{\frown}}{BAC}$ on a circle centered at point $D$, with points $A$, $B$, and $C$

1. **State the problem:** We have a circle centered at point $D$ with points $A$, $B$, and $C$ on it in clockwise order. We know the angles at the center: $\angle ADB = 7N + 12$, $

1. **State the problem:** We need to find the arc measure of the minor arc $\overset{\large{\frown}}{BC}$ in degrees. 2. **Identify given information:**

1. **State the problem:** We need to calculate the distance between two points A(50,32W) and B(50,32W) on the Earth. The Earth radius is given as 6400 km, and use $\pi=3.14$. 2. **

1. Stating the problem: We are given two points on Earth, \( A(50^{\circ}N, 32^{\circ}W) \) and \( B(50^{\circ}S, 32^{\circ}W) \), and need to find the distance between them. The E

1. **State the problem:** We want to find the total surface area of the region common to the three cylinders given by: $$x^2 + y^2 = 3^2$$

1. **State the problem:** We have two similar quadrilateral shapes, Q and R.

1. **Understand the problem:** We need to find the bearing from point C to point D on a square grid map. The bearing is the angle measured clockwise from the north direction to the

1. **State the problem:** Complete the sentence to describe the locus of points inside rectangle PQRS that are equidistant from sides PS and QR.

1. **State the problem:** We need to find the perimeter of a right-angled triangle with height 9 cm, base divided into segments of 2 cm and 12 cm, and the right angle between the h