📐 geometry

1. **Problem Statement:** We need to find:

1. **State the problem:** We need to find the area of a right triangle with legs measuring 6 cm and 7 cm. 2. **Formula:** The area $A$ of a triangle is given by

1. **State the problem:** We need to find the area of the given irregular shape composed of a rectangle and a right triangle. 2. **Identify the shapes and dimensions:**

1. **State the problem:** We are given the volume of a cone as $378\pi$ cm$^3$ and the radius $r=9$ cm. We need to find the missing height $h$. 2. **Formula:** The volume $V$ of a

1. **State the problem:** We need to find the radius $r$ of a cone given its volume and height. 2. **Given formula:** The volume $V$ of a cone is given by

1. **State the problem:** We have a triangle with sides 37 mm, 35 mm, and 38 mm, and one angle of 76°. We need to find the size of angle $\theta$ to 2 decimal places.

1. **State the problem:** We have three points A, B, and C. The bearing of B from A is 050° and the bearing of C from B is 110°. We need to find the size of angle ABC. 2. **Underst

1. **State the problem:** We need to determine the maximum occupancy for each room based on the area and the occupancy limit of 1 person per 1.7 square meters. 2. **Calculate the a

1. The problem asks for the formula to find the volume $V$ of a composite solid consisting of a hemisphere on top of a cone. 2. The volume of a hemisphere is given by the formula:

1. **Problem statement:** Given a circle with points A, B, C, and F on the circumference, and a tangent DE touching the circle at point B. The angle at F is 96° and the angle betwe

1. **Problem Statement:** We have two cylinders:

1. **State the problem:** Determine if polygons EFGH and WVUT are similar based on their given angles and side lengths. 2. **Recall similarity criteria for polygons:** Polygons are

1. **State the problem:** The user wants to solve for the values of $x$, $y$, and $z$ related to angles formed by chords intersecting inside a circle with a given 50° angle at the

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