📐 geometry

1. **Problem statement:** Find the length of side $BC$ in triangle $ABC$ where $AB=17$, $AC=8$, and there is a right angle at the foot of the altitude from $C$ to $AB$. 2. **Key co

1. **Problem statement:** We have triangle ABC with point D on segment AC such that BD is perpendicular to AC and BD bisects the angle at B. Given lengths are $AD=5$ and $BD=12$. W

1. **Problem Statement:** We have a right triangle $\triangle SRQ$ with a right angle at $R$. Point $T$ lies on segment $SQ$ such that $ST=9$ and $TQ=16$. The segment $RT$ is perpe

1. **Problem Statement:** We have an isosceles right triangle with an altitude drawn from the right angle vertex to the hypotenuse. The altitude length is $x$ units. We need to fin

1. **Problem Statement:** We have a right triangle WZX with a right angle at W. An altitude WY is drawn to the hypotenuse ZX, meeting at Y. Given WY = 4 units, ZY = 3 units, and we

1. **State the problem:** Find the area of the compound L-shaped figure made of rectangles with given side lengths. 2. **Identify the rectangles:** The figure can be divided into t

1. **State the problem:** We need to find the volume of a prism with given dimensions. 2. **Formula for volume of a prism:**

1. **State the problem:** We have a triangle SQT with angles $S=113^\circ$, $Q=4x$, and $T=6x+3$. We need to find the values of $x$ and the angles $Q$ and $T$. 2. **Use the triangl

1. **State the problem:** We are given two triangles with some angles expressed in terms of $x$ and some known angles. We need to write equations, solve for $x$, and find the missi

1. **Problem statement:** Calculate the angles corresponding to the decimal fractions in a semicircle diagram.

1. The problem involves a right triangle with sides named kateta (vertical leg), kateto (horizontal leg), and hipotenuza (hypotenuse). 2. According to the Pythagorean theorem, in a

1. **Problem statement:** Berechne den Abstand vom Bild zum Nagel, wenn das Bild mit einer 1,40 m langen Schnur an einem Nagel aufgehängt wird.

1. **State the problem:** Find the value of the missing interior angle $x$ in a triangle with angles 55°, 25°, and $x$°. 2. **Formula and rule:** The sum of interior angles in any

1. **Problem:** Find the value of $x$ in a triangle with interior angles $55^\circ$, $25^\circ$, and $x^\circ$. 2. **Formula:** The sum of interior angles in any triangle is always

1. **Problem Statement:** We have an 8-sided polygon (octagon) with a line segment from the center to the upper-left side measuring 14 units, and the right side of the polygon is l

1. The problem is to find the area of a regular pentagon with side length $8$ and apothem $5.5$. 2. The formula for the area $A$ of a regular polygon is:

1. **Problem 1:** Find the measure of each numbered angle (1, 2, 3) in the diamond-shaped quadrilateral with a given angle of 45°. 2. The diamond shape is a rhombus, so all sides a

1. **State the problem:** We need to find the total area of a kite made of two pieces of fabric sewn together at the midpoint of the base. 2. **Given data:**

1. **Problem 2(a): Write a congruency statement for the two triangles sharing line segment.** Given two triangles sharing a side, congruency can be stated by matching corresponding

1. **Problem statement:** Calculate the exact length of the unknown side in the right triangle with hypotenuse $\sqrt{85}$ and one leg $\sqrt{68}$. Then estimate the unknown base l

1. **State the problem:** We need to find the unknown side length in each right triangle using the Pythagorean theorem. 2. **Recall the Pythagorean theorem:** For a right triangle