📐 geometry

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:

1. **State the problem:** We have a cyclic quadrilateral PQRS inscribed in a circle with center O. Given: PR = 17.4 units, QR = 10.7 units, and \(\angle QPR = 21.7^\circ\). We need

1. **Problem Statement:** Prove the following about the circle and tangents:

1. **Problem Statement:** We are given a circle with points A, T, Q on its circumference. PT and PQ are tangents to the circle at T and P respectively. PK is parallel to chord QA,

1. **Problem statement:** We have a rectangle with sides 9 cm and 16 cm. We want to find the length of the diagonal $d$. 2. **Formula used:** The diagonal of a rectangle forms a ri

1. **State the problem:** We need to find the area of a triangle with a base of 30 inches and a height of 30 inches. 2. **Formula for the area of a triangle:**

1. **State the problem:** We need to find the area of a triangle with base $b = 21$ cm and height $h = 8$ cm. 2. **Formula for the area of a triangle:**

1. **State the problem:** We need to find the perimeter of a rectangle with vertices at approximately $(-6,6)$, $(-3,9)$, $(3,1)$, and $(-2,-2)$. 2. **Recall the perimeter formula

1. **State the problem:** We need to find the perimeter of a rectangle with vertices at approximately $(-3,0)$, $(-1,6)$, $(2,5)$, and $(0,-1)$. The perimeter is the total distance

1. **State the problem:** Find the distance between the points $(-4, -4)$ and $(6, -6)$ on the coordinate plane. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1

1. **State the problem:** We need to find the perimeter of a rectangle with vertices at approximately $(-2, -1)$, $(4, 1)$, $(5, -1)$, and $(-1, -4)$ on the coordinate plane. The r

1. **Problem Statement:** Determine the perimeter and area of the composite polygon with sides 5 in, 6 in, 7 in, 10 in, and 15 in.

1. **Problem statement:** Find the perimeter of the square with vertices at (0,0), (4,5), (0,9), and (-4,5). 2. **Formula:** The perimeter $P$ of a square is given by $P = 4s$, whe