📐 geometry

1. **Problem Statement:** Find the length of the hypotenuse $c$ in each right-angled triangle using the Pythagorean theorem. 2. **Formula:** The Pythagorean theorem states:

1. **Problem Statement:** Find the length of the hypotenuse $c$ in the right-angled triangle with legs 3 and 7, expressing the answer exactly using a surd. 2. **Formula:** Use the

1. **Problem:** Find the length of the hypotenuse $c$ in a right triangle given the legs. 2. **Formula:** Use the Pythagorean theorem:

1. **State the problem:** We need to find the area of each room in the floor plan, the total area to be carpeted, and the total cost of carpeting given the price per square meter.

1. **State the problem:** Calculate the total carpet area of the house using the given floor plan dimensions and then find the total carpet cost at 128.00 per square meter. 2. **Id

1. The problem asks if the triangles in question 11 are similar and if yes, how. 2. To determine similarity of triangles, we use the criteria:

1. **State the problem:** We have a rectangular prism with dimensions length $7$ cm, width $4$ cm, and height $4$ cm. A triangular pyramid (a tetrahedron) with vertical height $2$

1. **Stating the problem:** We have a right triangle with one angle of 45°. The side opposite the 45° angle is $3\sqrt{2}$, the hypotenuse is $x$, and the base is $y$.

1. **State the problem:** Given a right triangle ABC with right angle at C, and segment CE perpendicular to AB, creating two right triangles BEC (with angle B = 60°) and CEA (with

1. **State the problem:** We have a right triangle with one angle of 45°, the side adjacent to this angle is 10, the side opposite is $x$, and the hypotenuse is $y$. We need to fin

1. **State the problem:** We have a right triangle with a 60° angle, the side opposite this angle is 12, the adjacent side is $x$, and the hypotenuse is $y$. We need to find $x$ an

1. The problem asks to identify the relationship between angles 1 and 7 formed by two parallel lines $a$ and $b$ cut by a transversal $h$. 2. Important definitions:

1. **State the problem:** We are given two parallel lines $m$ and $n$ cut by a transversal $t$. Angle 7 measures $75^\circ$. We need to find which angle among 3, 5, 2, and 6 is NOT

1. **State the problem:** We have three parallel lines intersected by two transversals, creating segments with lengths 27, 15, 20, and a missing length $x$ on the bottom horizontal

1. **State the problem:** We need to find the area of a right-angled triangle with a base of 9 meters and a height of 8 meters. 2. **Formula:** The area $A$ of a triangle is given

1. **State the problem:** We have a 30°-60°-90° right triangle with hypotenuse length 10. We need to find the lengths of sides $x$ (opposite 60°) and $y$ (opposite 30°). 2. **Recal

1. **Stating the problem:** We have a right triangle with legs of lengths $x$ and 10, and hypotenuse $2x + 21$. We need to find the value of $x$. 2. **Formula used:** By the Pythag

1. **State the problem:** Find the equation of one median of the triangle with vertices A(2,3), B(8,1), and C(5,7), and find the point of concurrency (centroid) of the medians. 2.

1. **State the problem:** We need to find the lengths of the bases $AB$ and $DC$ of the isosceles trapezoid $ABCD$ to calculate its perimeter.

1. **State the problem:** We have a frustum formed by removing a smaller square pyramid from a larger similar square pyramid. The base edge length of the frustum is 36 mm, and the

1. **Problem statement:** We need to find the size of angle $k$ given that another angle in the figure is $81^\circ$. 2. **Assumption:** Since the problem involves angles and one a