📐 geometry

1. **Problem:** Two circles with radii 6 cm and 8 cm have centers 10 cm apart. Find the area common to both circles. 2. **Formula and explanation:** The area of intersection of two

1. **Problem:** Two circles with radii 6 cm and 8 cm have centers 10 cm apart. Find the area common to both circles. 2. **Formula:** The area of intersection of two circles with ra

1. **State the problem:** We are given two vertices of a triangle, $A(2,1)$ and $B(3,-2)$, and the area of the triangle is 5. The third vertex $C$ lies on the line $y = x + 3$. We

1. **Problem Statement:** We have two similar trapezoids. The larger trapezoid has a vertical side of length 8 and a bottom base of length 12. The smaller trapezoid has a vertical

1. **Problem Statement:** We have a square cardboard of side 80 cm. We cut out squares of side 10 cm from each corner to form an open box by folding the sides up. We want to find t

1. **State the problem:** We have a right triangle KLM with a right angle at L. Side KL = 47, side LM = 94, and we need to find the angle $x$ at vertex M. 2. **Identify the sides r

1. **Stating the problem:** We have a right triangle KLM with a right angle at L. The sides KL and LM are given as 47 and 94 respectively, and we need to find the angle $x^\circ$ a

1. **Stating the problem:** We need to prove that the segment (CA) is the perpendicular bisector of the segment [DB] in two different ways, and optionally find a third way.

1. **Problem:** Explain why opposite angles in a parallelogram are always congruent. 2. **Definition and properties:** A parallelogram is a quadrilateral with both pairs of opposit

1. **State the problem:** We need to find the volume of a prism with length 36 mm and a triangular cross-section with base 18 mm and height 29 mm. 2. **Formula for volume of a pris

1. **Problem statement:** We have a shape made from a cylinder and a hemisphere. The radius $r$ of both is 8 cm. The total volume is given as $\frac{2684}{3} \pi$ cm$^3$. We need t

1. **State the problem:** We have a triangle with three angles: $2x + 16$, $x + 1$, and $80$ degrees. We need to find the value of $x$. 2. **Recall the triangle angle sum rule:** T

1. **Problem:** Find the value of $x$ in a triangle with angles $(2x+16)^\circ$, $80^\circ$, and $(x+1)^\circ$. 2. **Formula:** The sum of angles in any triangle is always $180^\ci

1. **State the problem:** We have a square-based pyramid with base edge length $5.7$ cm and volume $162.45$ cm$^3$. We need to find (a) the area of its base and (b) its height. 2.

1. Let's start by stating the problem: We want to understand the concept of "distance" in geometry, which is the measure of how far apart two points are. 2. The formula to find the

1. **Stating the problem:** We are given two congruent figures with sides and angles labeled. We need to find the measure of angle $\angle N$ in one of the figures. 2. **Understand

1. **Problem Statement:** We are given two congruent quadrilaterals GHIJ and MNOP with various side lengths and angles. We need to find the measure of angle \(\angle M\) in quadril

1. **Problem Statement:** Given two congruent polygons JIHG and UVWX, find the length of side UV. 2. **Understanding Congruence:** Congruent polygons have corresponding sides and a

1. **Stating the problem:** We want to understand what direction ratios are in the context of vectors and lines in geometry. 2. **Definition:** Direction ratios are a set of three

1. Let's start by understanding **direction cosines**. Direction cosines are the cosines of the angles that a line makes with the coordinate axes in 3D geometry. 2. If a line makes

1. **Problem Statement:** We have a quadrilateral ABCD divided by diagonal AC into two triangles: ABD (Ram's land) and ADC (Sohan's land). We need to find which piece is larger and