📐 geometry

1. **State the problem:** We need to find the total surface area of a hemisphere with diameter 11 meters. 2. **Recall the formula:** The surface area of a sphere is given by $$4\pi

1. **State the problem:** We have a rectangular tank with length $3$ m and width $2$ m. It is filled with $2.7$ m$^3$ of water without overflowing. We need to find the height $h$ o

1. **Problem statement:** We have a star made of 6 identical quadrilaterals. Each quadrilateral has an angle $x$ at the star's points and two adjacent angles of $35^\circ$. We need

1. Problem: Find the missing side length $x$ in the right triangle with legs $16\sqrt{3}$ and $16$, and hypotenuse $x$. 2. Formula: Use the Pythagorean theorem for right triangles:

1. **Problem statement:** Calculate the bearing of point Y from point Z given the angles at points X and Y relative to north. 2. **Understanding bearings:** A bearing is measured c

1. The problem asks to verify the correctness of the given angle measures in two figures involving parallel lines and transversals. 2. For the first figure with angles a=83°, b=97°

1. The problem states that in parallelogram $ABCD$, angle $D$ is a right angle, and we want to prove that the diagonals are congruent. 2. Bao suggests proving this by using the fac

1. **State the problem:** Find the degree measure of the angle marked with an arc at the bottom of the figure, given that the angles are supplementary and some angles are labeled a

1. The problem asks to identify the location of the origin on the coordinate plane. 2. The origin in a coordinate plane is the point where the x-axis and y-axis intersect.

1. **State the problem:** We need to find the perimeter of a composite figure made of a rectangle and a semicircle attached to the right side of the rectangle. 2. **Identify given

1. **State the problem:** Determine if triangles TUS and GIH are similar. 2. **Recall similarity criteria:** Triangles are similar if their corresponding angles are equal or their

1. **State the problem:** We have two rectangles, one inside the other. The outer rectangle has dimensions 12 units by 9 units.

1. **State the problem:** We have two rectangles, one larger with dimensions 2 units by 8 units, and a smaller one inside it with dimensions 2 units by 3 units. We need to find the

1. **State the problem:** We have two rectangles stacked vertically. The top rectangle measures 2 units wide and 8 units tall. The bottom rectangle measures 2 units wide and 3 unit

1. **Problem Statement:** Find the area of the shaded region which is the inner rectangle framed by a 2-unit wide frame inside an outer rectangle of width 12 units and height 8 uni

1. The problem asks which congruence criterion applies to the two triangles sharing side $PQ$ with vertices $N, O, P$ (top triangle) and $N, Q, P$ (bottom triangle). 2. The top tri

1. **State the problem:** We have a triangle with two interior angles measuring 72° and 62°, and an exterior angle adjacent to the 62° angle labeled as $x^\circ$. We need to find t

1. **Stating the problem:** We have a pyramid (ostrosłup) and a plane parallel to its base that cuts the pyramid into two parts. We need to find the ratio of the volume of the smal

1. **State the problem:** Given that AC is the angle bisector of \(\angle BAD\), prove that \(\triangle ABC \cong \triangle ADC\). 2. **Identify given information:**

1. **Problem statement:** Find the length of segment $LK$ given the other segment lengths in the geometric figure. 2. **Given data:**

1. **Problem Statement:** We have a circle with center O, tangent FE at point E, and points D, E, F, G on or around the circle. Given that EG = GF and angle Ê₃ = x, we need to: