📐 geometry

1. **Stating the problem:** We have points P(3,8), Q(9,6), R(6,4) and a parallelogram APQR. The line AB intersects the y-axis at A and has equation $3y=2x+18$. We need to find the

1. The problem is to write a proof in columnar format, which typically involves two columns: one for statements and one for reasons. 2. The columnar proof format helps organize log

1. **Problem 23:** Given $MN \cong QO$, $MN \parallel QO$, $NR \perp MP$, and $OP \perp MP$, prove $NR \cong OP$. 2. Since $NR \perp MP$ and $OP \perp MP$, by definition of perpend

1. **State the problem:** We have trapezium ABCD with given sides AB=4 cm, AD=5 cm, BD=7 cm, and angle at C = 110°. We need to find (a)(i) angle \(\angle ABD\), (a)(ii) length BC,

1. **Problem 1:** Given an isosceles triangle ABC with area 3 cm² and angle ABC = 29°, find the length $x$ of the equal sides. 2. **Formula for area of a triangle:** $$\text{Area}

1. **State the problem:** Find the area of each triangle given two sides and the included angle. 2. **Formula:** The area of a triangle with two sides $a$ and $b$ and included angl

1. **Problem statement:** We have a sector OAB of a circle with center O and radius $R$ cm, with central angle $2\theta$ radians. A smaller circle with center C and radius $r$ cm t

1. Let's clarify the problem: You seem to be asking why a certain angle or value is not 88, and you mention that O2 should be 92 because of angles in a triangle. 2. The key rule he

1. **Problem Statement:** Calculate the size of angle $\hat{O}_1$ given that $O$ is the center of circle $HEATR$, $AOF$ is parallel to $EH$, $\hat{F}_2 = 60^\circ$, and $\hat{R}_1

1. **Problem statement:** We have a circle with center C and a chord of length 5.4 units subtending a 150° angle at the center. We need to show that the radius $r$ of the circle is

1. **State the problem:** We need to find the volume of a miniature globe with a diameter of 20 cm. 2. **Formula used:** The volume $V$ of a sphere is given by the formula:

1. **Stating the problem:** We have two parallel lines $l \parallel j$ cut by a transversal $m$. We need to find the values of angles $y$ and $x$ given that one angle is $67^\circ$

1. **State the problem:** We need to find the volume of a sphere with radius $r=9$ cm. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. **State the problem:** We need to find the volume of a sphere with radius $r=9$ cm. 2. **Formula:** The volume $V$ of a sphere is given by the formula:

1. **State the problem:** We are given two angles around point K: $\angle JKL = (12x + 3)^\circ$ and $\angle KLM = (6x - 3)^\circ$. We know $\angle JKL$ is a right angle, so it mea

1. **Problem statement:** We are given two angles at point R on a straight line: one angle measures $3x$ degrees and the other measures $9x$ degrees. We need to find the value of $

1. **Problem Statement:** We analyze translations and rotations of given figures on a coordinate plane.

1. **Stating the problem:** We are given several pairs of angles formed by rays originating from point C. We need to analyze these pairs based on the given geometric configuration.

1. The problem involves understanding the geometric description of a rectangular chip with a gold-colored pattern on a card. 2. The chip is rectangular with rounded corners, positi

1. **Problem 1:** Given $BD \perp AB$, $BD \perp DE$, and $BC \cong DC$, prove $\angle A \cong \angle E$. 2. **Problem 2:** Given $BC \cong DC$, $AC \cong EC$, prove $\triangle ABC

1. **Problem 1:** Given $BD \perp AB$, $BD \perp DE$, and $BC \cong DC$, prove $\angle A \cong \angle E$. 2. **Problem 2:** Given $BC \cong DC$, $AC \cong EC$, prove $\triangle ABC