📐 geometry

1. **State the problem:** We are given a triangle divided into two smaller right triangles with angles labeled as follows: one small triangle has an angle $3x - 5$, the other has a

1. Задача 10: Точките P и Q лежат на страните AC и BC на ΔABC с условия AP = \frac{1}{4}AC и CQ = \frac{1}{3}BC. Трябва да намерим вектора AB чрез векторите p = AP и q = CQ. 2. Изп

1. **Problem:** Find the area and perimeter of a rectangle with length 7 ft and width 5 ft. 2. **Formulas:**

1. **State the problem:** Prove that triangles $\triangle RTU$ and $\triangle RUS$ are similar. 2. **Given:**

1. **Problem Statement:** Identify which pairs of angles are consecutive interior angles given the parallel lines and transversal. 2. **Definition:** Consecutive interior angles ar

1. **Problem Statement:** Given two parallel lines $\overrightarrow{FH}$ and $\overrightarrow{IK}$ intersected by a transversal $\overrightarrow{EL}$ passing through points $G$ and

1. **Problem Statement:** We have a circle with two secants intersecting outside the circle at point H. One secant passes through points S and H, extending beyond H to F, with segm

1. **Problem Statement:** We have a circle with points L and G on the circumference. A chord FG passes through the circle and is intersected by line LS at point S. The chord FG is

1. **Problem 1:** Calculate the distance from the surface to the centre of the Earth given the Earth's circumference is 40000 km. 2. The formula relating circumference $C$ and radi

1. **Problem statement:** Calculate the radius or diameter of a circle given its circumference, correct to three significant figures. 2. **Formula:** The circumference $C$ of a cir

1. **Problem statement:** Find the area and perimeter of a rhombus with diagonals of lengths 30 cm and 16 cm. 2. **Formulas:**

1. **Problem statement:** We have a pyramid with rectangular base ABCD where $AB=20$ cm, $BC=16$ cm, and the vertex $E$ such that $AE=BE=CE=DE=17$ cm. We need to find the angle bet

1. **Problem statement:** Given points A(7,7), B(8,4), and C(6,0) lie on the same circle, find: a) the equation of the perpendicular bisector of AB

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