📐 geometry

1. **State the problem:** We have two parallel lines $a$ and $b$ cut by a transversal $t$. Given $m\angle 3 = 4x - 31$ and $m\angle 8 = 2x + 7$, find the value of $x$ (Question 14)

1. **Stating the problem:** We have a circle with a tangent line at point F and a radius FG. The angle between the tangent line at F and the chord FG is given as 67°. We need to fi

1. **State the problem:** We need to find the volume of a composite shape made of a cone and a hemisphere. The cone has height $h=15$ cm and radius $r=7$ cm. The hemisphere has rad

1. **State the problem:** We have triangle DEF formed by connecting the midpoints of the sides of triangle ABC. We know the side lengths of triangle DEF are 2, 3, and 4. We need to

1. **Problem Statement:** We need to find a sequence of rigid transformations and dilations that maps square ABCD (side length 5) to square EFGH (side length 2).

1. **Problem Statement:** We are given two triangles, ABC and AED, with side lengths AB=8, BC=4, AC=6, and AE=4, ED=2, AD=3. We need to write two different sequences of transformat

1. The problem asks for the measure of angle A' B' C' after a dilation of right angle ABC with center P and scale factor $\frac{1}{2}$. 2. Important rule: Dilation changes the size

1. **Problem 1: Find a sequence of rigid motions and dilations that takes square ABCD to square EFGH.** - Given: Square ABCD with side length 5.

1. **Problem statement:** Given parallelogram RSTU, find the length of diagonal SU. 2. **Key property:** In a parallelogram, the diagonals bisect each other. This means the point V

1. **State the problem:** We have a parallelogram PQRS with segments on its diagonals given as $PQ = x + 29$, $QR = 7x + 5$, $QP = 3y$, and $PS = 8y - 35$. We need to find the valu

1. **State the problem:** Calculate the volume of a house-shaped figure composed of a rectangular prism base and a triangular prism roof. 2. **Given dimensions:**

1. **State the problem:** Calculate the volume of a trapezoidal prism with given dimensions: height of vertical face = 20 cm, base length = 45 cm, base depth = 17 cm, slant height

1. **State the problem:** Calculate the volume of a 3D house-like figure composed of a rectangular prism base and a triangular prism roof. 2. **Identify the shapes and dimensions:*

1. **State the problem:** We have a rectangular prism with dimensions 3 cm, 2 cm, and 10 cm. We want to find the volume if all dimensions are converted to inches. 2. **Formula for

1. **Problem Statement:** Verify the properties of quadrilateral KLMN with vertices K(-1, 4), L(2, 2), M(0, -1), and N(-3, 1): a) Check if it is a square.

1. **Problem statement:** Verify that the quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS with vertices P(0, 6), Q(-6, -2), R(2, -4), and S(4, 2) i

1. **Problem:** Verify that the diagonals of quadrilateral ABCD with vertices A(0,0), B(2,3), C(5,1), and D(3,-2) are perpendicular. 2. **Formula and rules:**

1. The problem asks for the length of the hypotenuse of a right triangle with legs measuring 11 cm and 10 cm. 2. We use the Pythagorean theorem, which states that in a right triang

1. The problem asks for the simplest way to find angle $Y$ in triangle $XYZ$ where $\angle X = 40^\circ$, $\angle Z = 60^\circ$, and side $XZ = 12.0$ cm. 2. Recall that the sum of

1. **Stating the problem:** We are given a triangle with angles $43^\circ$, $100^\circ$, and $37^\circ$ at vertices A, B, and C respectively, and side lengths $AB=3$ cm, $AC=7$ cm,

1. The problem is to understand the definition and relationship between two lines in geometry. 2. Two lines can be related in several ways: they can be parallel, intersecting, or c