📐 geometry

1. The user's message mentions volume of cones and pyramids but provides no specific problem or numerical data. 2. Without a clear problem statement or given values, we cannot calc

1. **State the problem:** Understanding the properties of inscribed angles and intercepted arcs in a circle based on given theorems and angle measures. 2. **Theorem 1:** The measur

1. **State the problem:** In parallelogram BIRD, with point G on triangle GIR such that IG bisects \( \angle BIR \), prove that \( \frac{BE}{EI} = \frac{RG}{GI} \). 2. **Recall pro

1. Problem statement: We have a circle with center O and radius OS = 7 cm. R is the midpoint of OS, so OR = RS = $\frac{7}{2} = 3.5$ cm. We have two sectors ORU (40°) and OST (50°)

1. We need to find the area of the unshaded region. 2. To do this, first identify the total area of the figure (usually a rectangle, square, or circle).

1. Given: OS = 7 cm and R is the midpoint of OS, so OR = RS = 3.5 cm. 2. (a) Diameter of OS: Since OS is a radius, diameter = 2 \times OS = 2 \times 7 = 14 cm.

1. Solve for the unknown values with explanations: a) Given angles 40°, 5x, and 65° between two parallel vertical lines with a transversal, these three angles form a straight line,

1. نبدأ بكتابة معادلة المستقيم: $$2y + x = 0$$

1. The problem states that triangle ABC is congruent to triangle PQR and that \( \angle BAC = \angle QPR \). We are asked to find which side in triangle PQR is equal to side BC in

1. Solve for the unknown values with reasons: **a)** Given angles 40°, 5x, and 65° produced by two parallel lines and a transversal.

1. The problem describes a right triangle with legs along the axes: one leg on the x-axis of length $3$, and the other leg on the y-axis of length $4$. 2. We are asked to find the

1. **Problem statement:** We have a circle with center $T$, $FR$ as diameter, with $FI \perp FM$, $\angle RFM = 30^\circ$, and $\angle IFB = 40^\circ$. We want to find $\angle BFR$

1. The problem states that ABC and FBD are straight lines with $x + y = 142^\circ$. We need to find $x$, $y$, and $z$. \n\n2. Since ABC is a straight line, the angles on it must su

1. **State the problem:** We are given a triangle ABC with BC = $5\sqrt{2}$ cm, \(\angle ABC = 30^\circ\), and \(\sin(\angle BAC) = \sqrt{5}/8\). We need to find the length of AC i

1. **State the problem:** We have a circle centered at point O. Line segment \(\overleftrightarrow{AC}\) is tangent at point C.

1. Stating the problem: Given a triangle ABC with perimeter 15 cm, side AB = 7 cm, and angle BAC = 60 degrees, find the lengths of AC and BC and the area of the triangle. 2. Define

1. Stating the problem: We need to find the volume of water in a cuboid container. The container has dimensions length $16$ cm, width $7$ cm, and height $14$ cm, but water only fil

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