📐 geometry

1. **Problem Statement:** Calculate the value of $x$ to 4 significant figures given two circles each of radius 10.5 cm with centers $A$ and $B$ touching at $T$. Given angles $\angl

1. **Problem:** Find the unknown angle in a triangle with angles 65°, 57°, and ?. 2. **Formula:** The sum of angles in any triangle is always 180°. That is,

1. **Problem statement:** Given a triangle with sides $a$, $b$, and $c$, and the relation $$\frac{1}{a+b} + \frac{1}{b+c} = \frac{3}{a+b+c},$$ find the angle opposite side $b$. 2.

1. **State the problem:** We need to find the volume of a shape made from a hemisphere and a half-cylinder. The hemisphere has radius $r=5.3$ cm, and it sits on the half-cylinder w

1. **Problem:** The Ferris wheel has a radius of 20 m. The platform is tangent to the Ferris wheel. How far is the platform from the center of the wheel? 2. **Formula and Explanati

1. **Problem statement:** Calculate the length $x$ in the triangle where $MN \parallel BC$ and given lengths are $BM=5$, $AN=4$, and $BC=10$. 2. **Formula and theorem used:** Accor

1. **Énoncé du problème :** Montrer que le point $G$ est le barycentre des points $(D;1)$ et $(H;y)$. 2. **Rappel de la définition du barycentre :** Le barycentre $G$ des points $D

1. **Problem statement:** We want to find how a line divides a sector into two equal areas. 2. **Understanding the sector:** A sector is a portion of a circle bounded by two radii

1. The problem states that the area of triangle $ADE$ equals the area of quadrilateral $DEBC$, and both are half the area of triangle $ABC$. It is given that this half area equals

1. **Problem statement:** We have a sector ABC of a circle with radius $a$ and central angle $\angle BAC = \frac{\pi}{6}$. Points D and E lie on AB and AC such that $AD = AE = ka$

1. The problem involves understanding a 3D block pyramid made of 1 cm x 1 cm x 1 cm cubes, where the base layer has 9 cubes in a row, and each upper layer decreases by one cube unt

1. **Problem statement:** Jon is tiling a 2-foot wide border around a bathtub that measures 5 feet by 3 feet. Each tile measures 4 inches by 4 inches. We need to find the minimum n

1. **Problem statement:** Jon is tiling a 2-foot wide border around a bathtub that measures 5 feet by 3 feet. Each tile measures 4 inches by 4 inches. We need to find the minimum n

1. **State the problem:** We need to find the number of sides $n$ of a regular polygon whose interior angle is $150^\circ$. 2. **Formula used:** The measure of each interior angle

1. **Complementary Angles** A complementary angle pair sums to 90°.

1. **Problem Statement:** We have a kite-shaped quadrilateral with angles labeled 1 through 7 and some given angle measures: $\angle 1 = 73^\circ$ and $\angle 5 = 46^\circ$. We nee

1. **Problem Statement:** We are given a kite-shaped quadrilateral EFGD with diagonals intersecting at H. The angle at vertex D, specifically $\angle GDE$, measures 59°. We need to

1. **Problem 1: Complementary Angles** A right angle measures 90°. Two angles are complementary if their sum is 90°.

1. **State the problem:** We need to find the missing angle $\angle U$ in triangle $UVT$ where $\angle V = 50^\circ$ and the exterior angle at $T$ (formed by extending $TP$) is $11

1. **Problem Statement:** We are given two intersecting lines forming four angles. One angle is 39°, the angle opposite it is unknown (?), and two other angles in a triangle are 20

1. **State the problem:** We have two triangles side by side. The left triangle has angles 60° and 65°, and the right triangle has angles 35° and 50°, with the third angle unknown.