📐 geometry

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

1. **Problem Statement:** Find the ratio between the area of triangle $\triangle ABC$ and $\triangle DEF$ given their side lengths. 2. **Given:**

1. The problem is to find the measure of angle ADC given that angle ADC is 134° and the answer is 88°. 2. This seems to be a misunderstanding or incomplete information because angl

1. **Problem statement:** In circle ABC passing through center O of circle ADC, with ADB a straight line and given \(\angle BDC = 46^\circ\), find \(\angle CBD\). 2. **Key facts an

1. **Problem statement:** We are given a triangle ABC with sides AC = 130, side from C to the unlabeled vertex = 110, and an angle of 38° at the unlabeled vertex adjacent to the si

1. **State the problem:** We have a triangle ABC with side AC = 3, angle A = 35°, and angle C = 85°. We need to find the length of side BC, denoted as $x$. 2. **Use the Law of Sine

1. **Problem statement:** In triangle $\triangle ABC$, angles are given as $\angle A = 30^\circ$, $\angle B = 90^\circ$, and $\angle C = 60^\circ$. Point $X$ lies inside $\triangle

1. **State the problem:** We have a pentagon with five sides and four known angles: 50°, 120°, 80°, and one unknown angle marked with a question mark (?). We need to find the value

1. **Problem statement:** We are given a triangle with side BC = 376, angle A = 98.4°, and angle B = 24.6°. We need to find side AB = x using the Law of Sines. 2. **Law of Sines fo

1. **State the problem:** We are given a triangle ABC with angles $A=98.4^\circ$, $B=24.6^\circ$, and side $c=376$ opposite angle $C$. We need to find side $x$ opposite angle $B$ u

1. **Problem Statement:** Given circle ⊙A with points and angles: $m\angle LAM=42^\circ$, $m\angle HAG=30^\circ$, and $\angle KAH$ is a right angle ($90^\circ$). Find the measures

1. The problem states that ABCD is a rectangle with side lengths AD and AB in the ratio 2:4, and the length DC is given as 16 cm. 2. Since ABCD is a rectangle, opposite sides are e

1. **Problem Statement:** A solid toy is made by surmounting a right circular cone on a hemisphere. Both have the same radius $r = 7$ cm. The height of the cone is $h = 29$ cm. Fin

1. State the problem: You want to find the volume of a sphere given its radius. 2. Use the formula for the volume of a sphere: $$V = \frac{4}{3} \pi r^3$$