📐 geometry

1. **Problem Statement:** We have a cyclic quadrilateral ABCD inscribed in a circle with given sides: $AB=22$, $BC=35$, $CD=24$, and diagonal $AC=40$. We need to find:

1. **Problem statement:** Given triangle ABC with sides $a=7$ cm, $c=5$ cm, the distance from the incentre to side BC is 1.477 cm, and the area is 16.25 cm², find: - (a) side $b$

1. The problem asks for the compass heading that is 27.5° south of west. 2. Compass headings are measured clockwise from north (0°).

1. **State the problem:** We need to find the value of $a$ given the angles in a triangle where one angle is $90^\circ$, another is $21^\circ$, and the third is $(2a + 6)^\circ$. 2

1. **State the problem:** We need to find the length of side $x$ (WU) in right triangle WUV, where the right angle is at $V$, side $WV = 9.2$, and angle $U = 50^\circ$. 2. **Identi

1. **State the problem:** We are given angles in a triangle and need to find the value of $a$. 2. **Given angles:** $(2a + 6)^\circ$, $21^\circ$, $90^\circ$, $90^\circ$, and $a^\ci

1. **State the problem:** We need to find the area of quadrilateral EFGH with given side lengths: EH = 5 cm (vertical), GH = 10 cm (horizontal), FG = 2 cm (vertical), and EF is dia

1. **State the problem:** We need to find the perimeter of a right triangle with a vertical height of 9 cm and a base divided into two segments of 4 cm and 12 cm. 2. **Identify the

1. **Énoncé du problème :** Nous avons un triangle $ABC$ et un point $D$ tel que $\overrightarrow{CD} = \overrightarrow{AC} + 2 \overrightarrow{BC}$. On définit $D_1$ comme le proj

1. **Problem Statement:** We have two rectangles. The larger rectangle has an area of $170$ square units and a height of $34$ units. The smaller rectangle has an area of $102$ squa

1. **Problem Statement:** Given triangle $ABC$ with points $A(4,3)$, $B(-1,-1)$, and $C(2,2)$, find:

1. **Problem Statement:** We are given an isosceles triangle \(\triangle FOR\) with \(\overline{OR} \parallel \overline{AN}\). We know the lengths \(FO = 7.0\), \(OR = 11.2\), and

1. **Problem statement:** We have a square with side length $7.4$ cm. A curved line composed of two circular arcs connects points $A$ and $B$, which lie at the midpoints of the lef

1. **State the problem:** We need to find the area of the shaded quarter circle inside each square. 2. **Formula:** The area of a quarter circle is given by $$\text{Area} = \frac{1

1. **Problem Statement:** Find the unknown angle $x$ in each polygon given the other interior angles. 2. **Formula:** The sum of interior angles of a polygon with $n$ sides is give

1. **Problem 1: Area of the ring-shaped shaded part** We have two circles: a large circle with radius $6$ m and a smaller inner circle with radius $4.1$ m. The shaded area is the a

1. **State the problem:** Calculate the circumference of a trundle wheel with diameter 40 cm.

1. The problem is to find the length $x$ of the hypotenuse in each right triangle given the legs. 2. We use the Pythagorean theorem: $$x = \sqrt{a^2 + b^2}$$ where $a$ and $b$ are

1. **Stating the problem:** We need to find the length of the side $CD$ of the quadrilateral $ABCD$ given the coordinates of points $C$ and $D$.

1. **Problem statement:** Given a pyramid $S.ABCD$ with base $ABCD$ a trapezoid where $AD = 2BC$. Points $E, F, Q$ are midpoints of edges $SA, SD, AB$ respectively. (a) Prove that

1. **Find the equation of the line of intersection of the planes:** Given planes: