📐 geometry

1. Problem: If line M and N are both perpendicular to line S, then M and N are said to be ___. - When two lines are both perpendicular to the same line, they are parallel to each o

1. **Problem:** Prove that in a quadrilateral ABCD circumscribing a circle, the sum of the lengths of opposite sides are equal, i.e., $AB + CD = AD + BC$. 2. **Formula and rule:**

1. Problem 1: A circle is inscribed in a square of side 10 cm. Find the area of the circle in terms of $\pi$. - The diameter of the inscribed circle equals the side of the square,

1. **Problem Statement:** Find the area of the grassy border around a circular fountain with radius 5 m, where the grass border extends to 8 m from the center.

1. Let's state the problem: We have a rectangle with height 20, a trapezium with height 15 mm, and a parallelogram with height 15 mm. 2. To find areas of these shapes, we use the f

1. **Stating the problem:** We have a metal strip of length 150 mm and height 20 mm. From this strip, three congruent trapeziums and two congruent parallelograms are pressed out. W

1. Let's start by defining the shapes: a trapezoid is a quadrilateral with at least one pair of parallel sides, and a parallelogram is a quadrilateral with two pairs of parallel si

1. **Problem Statement:** Given that AD and BC are equal in length and both are perpendicular to the line segment AB, prove that the line segment CD bisects AB. 2. **Understanding

1. **Problem Statement:** Given that AD and BC are equal in length and both are perpendicular to the line segment AB, prove that the line segment CD bisects AB. 2. **Understanding

1. **Problem Statement:** We have a metal strip composed of five trapezoidal shapes aligned horizontally with a total length of 150 mm and height 20 mm. We need to find the area of

1. **State the order of rotation and angle of rotation for each figure:** - The order of rotation is the number of times a figure maps onto itself during a full 360° rotation.

1. **Problem statement:** Given a right triangle ABC with a right angle at A, AC = 27 cm, BC = 45 cm, and AB = 3x cm. Find the value of $x$. 2. **Formula used:** In a right triangl

1. The problem asks us to find the coordinates of the vertices E, F, G, and H after a 180° counterclockwise rotation about the origin. 2. The original coordinates are:

1. **Problem statement:** Given a right triangle ABC with right angle at A, AC = 27 cm, BC = 45 cm, and AB = 3x cm, find the value of $x$. 2. **Formula used:** Pythagorean theorem:

1. **Problem Statement:** We are given a triangle \(\triangle PQR\) with vertices \(P(5,6)\), \(Q(6,7)\), and \(R(4,8)\). We want to find the coordinates of the image of \(\triangl

1. **Problem Statement:** We are given a triangle PQR with vertices at points $P(3,6)$, $Q(4,8)$, and $R(2,9)$. We need to find the coordinates of the triangle after translating it

1. **Stating the problem:** We need to find the volume of a trapezoidal prism with given side lengths. 2. **Formula for volume of a prism:**

1. Let's identify the problem: We need to find the missing lengths of two triangles, one purple and one rusty looking, presumably right triangles. 2. The key formula for right tria

1. **Problem Statement:** Find the hypotenuse of the top right-angled triangle with legs 5 cm and 5 cm. Also, find all missing lengths of the stacked right-angled triangles from bo

1. **Problem statement:** We are given points $P(1,1)$ and $Q(7,11)$ which form the diameter of a circle.

1. **Problem statement:** We have two circles: a smaller circle with center A and radius $r$ passing through points B, C, and D, and a larger circle with center C and radius $s$ pa