📐 geometry

1. Problem (i): Prove that a point O equidistant from points A and B lies on the perpendicular bisector of segment AB. 2. Recall the definition: The perpendicular bisector of a seg

1. **Stating the problem:** We need to find the difference between the total green shaded area and the total blue shaded area on a grid with a scale in centimeters. 2. **Understand

1. The problem involves understanding the relationship between the angles and sides of two parallelograms in a tiled pattern. 2. Key properties of parallelograms:

1. **Problem Statement:** We have a mechanical jack shaped as a parallelogram with all four arms of equal length (a rhombus). The crank is 5 inches above the ground, and we want to

1. **Problem Statement:** We have a mechanical jack shaped as a parallelogram with all four arms of equal length. The crank is 5 inches above the ground, and we want to find the he

1. **State the problem:** Given a parallelogram ABCD with AB \parallel DC and AD \parallel BC, prove that opposite angles are congruent: \(\angle A \cong \angle C\) and \(\angle B

1. **Problem Statement:** If a quadrilateral is a parallelogram, prove that its opposite sides are congruent and its diagonals bisect each other.

1. **State the problem:** We are given a parallelogram with sides labeled as follows: top side = $12 - c$, left side = $2b$, bottom side = $c - 2$, right side = $4b - 6$. We need t

1. **Stating the problem:** We have two adjacent parallelograms sharing a bottom angle, with the top left angle of the upper parallelogram given as $41^\circ$ and the top right ang

1. **Problem:** How many triangles can you identify in the given figure where a square is divided by its diagonals into four triangles, then further divided by a vertical line thro

1. **Problem statement:** We have a six-pointed star made up of 6 identical quadrilaterals. Each quadrilateral has angles marked $x$ and two angles marked $25^\circ$. We need to fi

1. **Stating the problem:** We have a horizontal line segment AB of length 8 cm.

1. **State the problem:** Calculate the volume of a shape made from a triangular prism with a rectangular hole through the middle. 2. **Identify the dimensions:**

1. The problem involves identifying congruent angles and setting up a proportion to show that two triangles are similar. 2. From the given labels, Angle C is congruent to Angle A b

1. **State the problem:** We need to find the volume of a cylinder with radius $r=12.8$ m and length (height) $h=5.4$ m. 2. **Formula:** The volume $V$ of a cylinder is given by th

1. **State the problem:** Calculate the cross-sectional area and volume of a cylinder with radius $r=15$ cm and height $h=27$ cm.

1. The term is **perpendicular bisector**: a line, ray, or segment that divides a segment into two congruent parts at a 90° angle. 2. The term is **angle bisector**: a line, ray, o

1. **State the problem:** We have two intersecting lines forming four angles, with one angle given as 80° and its vertically opposite angle labeled as $y$. Below the intersecting l

1. **State the problem:** We are given a trapezium with area $6\sqrt{21}$ cm², top base $5$ cm, height $\sqrt{7}$ cm, and bottom base $f$ cm. We need to find $f$ in the form $x\sqr

1. **Estimate the area of the pond by adding the areas of two trapezoids.** The formula for the area of a trapezoid is:

1. **Problem:** Michael drew a scale drawing where 0.5 inches equals 1 foot. a. If the hallway is 2.5 inches wide in the drawing, find the actual width.