📐 geometry

1. **Problem statement:** We have a right-angled triangle XYZ with the right angle at Y. Side XY measures 5 cm, side XZ (the hypotenuse) measures 19 cm, and we need to find the len

1. The problem states the Pythagorean Theorem: $$a^2 + b^2 = c^2$$ and gives values $a=7$ and $b=11$. We need to find $c^2$. 2. First, calculate $a^2$ and $b^2$:

1. **Problem statement:** Prove that the straight line joining the origin to the intersection of the line $kx + hy = 2hk$ and the circle $(x - h)^2 + (y - k)^2 = a^2$ is perpendicu

1. The problem involves understanding the dimensions of a geometric shape with given lengths: total width 210 units, height 170 units, a 65-unit segment from the left to a notch, a

1. **Stating the problem:** We have two pieces, each a rectangle of dimensions 6 cm by 2 cm, and we want to arrange them to form a figure with a perimeter of 22 cm. 2. **Understand

1. **Problem statement:** A solid toy is made by mounting a cone on top of a hemisphere. The height of the cone is 7 cm and the radius of both the cone and hemisphere is 3.5 cm. We

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.