📐 geometry

1. **Problem 9:** Find $x$ if $KD = x + 7$ and $KN = 2x - 4$. Since $K$ is the centroid, it divides the median into a ratio of 2:1, so $KD = 2 imes KN$.

1. **State the problem:** We have a circle with center O and two chords intersecting at O forming a 50° angle. We want to find the values of $x$, $y$, and $z$ which are angles rela

1. **State the problem:** We are given a right triangle GHF with right angle at G. Given sides:

1. **State the problem:** We are given a right triangle MEJ (labeled KIJ in the diagram) with a right angle at vertex I. Given sides:

1. **State the problem:** We need to find the length of segment $\overline{AB}$. Given the triangle with points $A, B, C, D, E$ and side lengths $BC=3$, $CD=4$, and $DE=12$. 2. **A

1. **State the problem:** Verify that triangles \(\triangle ABC\) and \(\triangle DEF\) are similar and find the scale factor from \(\triangle ABC\) to \(\triangle DEF\). 2. **Give

1. **Problem statement:** We have a triangle with one angle measuring 116° and two sides adjacent to this angle measuring 2.7 cm and 4.9 cm. We need to find the unknown angle $x$ o

1. **State the problem:** We are given a regular convex polygon where each interior angle measures 144°. We need to find how many sides the polygon has. 2. **Formula for interior a

1. The problem is to state the Circle Theorem. 2. The Circle Theorem states that the angle subtended by a diameter of a circle at any point on the circle is a right angle (90 degre

1. **Problem:** Calculate the unknown side of the triangle with sides 35 cm and 37 cm. 2. **Formula:** Use the Pythagorean theorem for right triangles: $$c^2 = a^2 + b^2$$ where $c

1. **State the problem:** We need to find the measure of the smallest angle in a triangle with sides 12, 23, and 34, where the smallest angle is opposite the side of length 12. 2.

1. **State the problem:** We need to find the measure of the largest angle in a triangle with sides 12, 23, and 34. The largest angle is opposite the longest side, which is 34. 2.

1. **State the problem:** Find the distance between the two parallel planes given by the equations: $$z=2y-2x$$

1. **State the problem:** We need to find the length of the arc AB of a sector of a circle with radius 43 mm and central angle 62°. 2. **Formula for arc length:** The length of an

1. **State the problem:** We need to find the surface area of a birdhouse with a rectangular base, a triangular prism roof, and a circular hole on the front triangular face.

1. **State the problem:** Determine the surface area of the birdhouse with a rectangular base of dimensions 26 cm by 25 cm, a height of 18 cm, and a triangular roof with slant heig

1. **Problem statement:** Given rays \(\overrightarrow{OC}\), \(\overrightarrow{OD}\), \(\overrightarrow{OE}\), \(\overrightarrow{OF}\), and \(\overrightarrow{OG}\) bisecting angle

1. **Problem statement:** Find the length of segment $DE$ given the geometric configuration with points $A, B, C, D, E, F, G$ and the following data: $AG=3$, $GB=7$, angles at $C$

1. **Problem 1: Area of parallelogram B** We are given two parallelograms A and B with bases 4 cm and 6 cm respectively.

1. **Problem:** Two similar triangles have bases 7 cm and 14 cm. Triangle A has area 20 cm². Find the area of triangle B. 2. **Formula:** Areas of similar triangles scale as the sq

1. **Problem 1: Areas of Similar Triangles** We have two similar triangles A and B.