📐 geometry

1. **Problem statement:** We have a right triangle with a hypotenuse of length $13\sqrt{2}$, angles $45^\circ$ and $30^\circ$, and a perpendicular dropped from the top vertex to th

1. **State the problem:** We have a rectangular wall divided into three sections: red, green, and blue. The green section is a square with side length 14 feet.

1. **Problem statement:** Find the missing side length of each rectangular solid given the volume. 2. **Formula:** Volume of a rectangular solid is given by $$V = l \times w \times

1. The problem asks to identify points from ordered pairs and vice versa, and to plot points on a coordinate grid. 2. An ordered pair is written as $(x,y)$ where $x$ is the horizon

1. **Problem statement:** Given triangle RST with sides RS = 4, RT = 3, and ST = 5, show that triangle RST is a right triangle. 2. **Formula used:** To verify if a triangle is righ

1. **State the problem:** Determine the type of triangle based on the side lengths and verify if it is right, acute, or obtuse. 2. **Recall the Pythagorean theorem:** For a triangl

1. **State the problem:** We are given a straight horizontal line intersected by two diagonal lines forming several angles. The angles given are $y^\circ$ and $55^\circ$ on the lef

1. **Problem statement:** Given triangle ABC with sides AC = 26 inches, BC = 20 inches, and angle ABC = 95°, find the angle $\theta$ at vertex C. 2. **Formula used:** Use the Law o

1. The problem asks to identify which shortcut can be used to prove that the given pair of triangles is congruent. 2. The common shortcuts for triangle congruence are:

1. **Problem statement:** Bestem radius i cirklerne der indeholder buerne BC og FG, givet at BC og FG er cirkelbuer, og at linjerne AB, CD, EF og GH er rette linjestykker.

1. The problem asks to select the shortcut to prove two right triangles are congruent. 2. The triangles share a side and both have right angles, so each has a right angle.

1. **Problem statement:** Calculate the perimeter of the figure in problem 16, which consists of three semicircles placed horizontally. The largest semicircle has a diameter of 20

1. **State the problem:** Find the perimeter of sector AOB of a circle with radius 18 cm and central angle 80°. 2. **Recall the formula for arc length:**

1. **State the problem:** We are given a triangle with one angle measuring 75°, another angle unknown (labeled "?"), and an exterior angle adjacent to the unknown angle measuring 1

1. **State the problem:** Determine which transformation takes Figure A to Figure B based on their vertex coordinates. 2. **List the vertices:**

1. **Problem statement:** Given two parallel lines $c$ and $d$ cut by transversals $a$ and $b$, with $m\angle 12 = 60^\circ$ and $m\angle 14 = 80^\circ$, find $m\angle 3$ and $m\an

1. **State the problem:** We are given a quadrilateral EBCD with angles at points B, C, and F. We know that $m\angle B = (14x + 4)^\circ$, $m\angle C = 51^\circ$, and $m\angle F =

1. Problem: Find the area of circles given the circumference. 2. Formula: Circumference $C = 2\pi r$ and Area $A = \pi r^2$.

1. **State the problem:** Find the coordinates of the orthocenter of the triangle with vertices at points $A(0,0)$, $B(8,2)$, and $C(2,3)$. The orthocenter is the point where the t

1. **State the problem:** We have three right triangles arranged horizontally. The first triangle has legs 8 (vertical) and 6 (horizontal). The second triangle shares a vertical le

1. The problem asks for the volume of a sphere with a diameter of 4 inches. 2. The formula for the volume of a sphere is $$V = \frac{4}{3} \pi r^3$$ where $r$ is the radius.