📐 geometry

1. **Problem statement:** Given a circular body with intersecting chords in a quadrilateral shape, with points A, B, C, D, and E inside the figure.

1. **State the problem:** We need to find the perimeter of a right-angled triangle with one angle of 43° and the side adjacent to this angle (base) measuring 13.8 m. 2. **Identify

1. **Problem statement:** Given a stadium-shaped figure with rectangle length $AB = CD = 100$ m, width $BC = AD = 40$ m, and diagonal $AC = 118.7$ m, find $\sin(\angle ACD)$, $\cos

1. **Problem statement:** We need to find the height $y$ of an isosceles triangle where the two equal sides each measure 11.5 m and the angle adjacent to the height is 57°. 2. **Un

1. **State the problem:** We need to find the area of a rectangular rug with width $8$ m and height $2 \frac{7}{12}$ m. 2. **Formula:** The area $A$ of a rectangle is given by:

1. The problem states that point Q is the midpoint of line PR, and we are given points P(-4, 6) and Q(k, -2). We need to find the value of $k$ given that the distance $PQ = 17$ uni

1. **Problem statement:** We have a right-angled triangle and an isosceles triangle joined together. The equal sides of the isosceles triangle are 25 cm each, and the base of the r

1. **Problem Statement:** Prove the projection law in any triangle $\triangle ABC$ using vector methods: $$a = b \cos C + c \cos B$$ where $a$, $b$, and $c$ are the sides opposite

1. **Problem statement:** In the figure, AB \parallel CE and DF \parallel AC. Given angles are \(\angle BAC = 71^\circ\), \(\angle EFC = 80^\circ\), and \(\angle BFE = 4x^\circ\).

1. **Problem statement:** Given that lines AB // CD and BP // AD, with angles \(\angle EBF = 68^\circ\) and \(\angle CDE = 58^\circ\), find (i) \(\angle AEB\) and (ii) \(\angle ABE

1. Let's clarify the step where we calculate $180 \times 5 = 900$.\n\n2. This step comes from the formula for the sum of the interior angles of a polygon: $$\text{Sum of interior a

1. **State the problem:** We have a 7-sided polygon (heptagon) with known angles 165, 150, 123, 100, 170, and two unknown angles labeled $x+11$ and $x$. We want to find the value o

1. **Stating the problem:** We have a 7-sided polygon (heptagon) with sides 165, 150, 123, 100, 170, and two unknown sides labeled $x+11$ and $x$. We want to find the value of $x$.

1. **State the problem:** We have a composite object made of two cylinders stacked vertically. The top cylinder has diameter 4 cm and height 14 cm, and the bottom cylinder has diam

1. The first problem involves determining the area of a figure given side lengths 14 cm, 4 cm, and 12 cm, with multiple choice answers for the area in cm². 2. The second problem as

1. **Problem 1:** Find the surface area of a prism 55 cm long with square ends. 2. The prism has square ends, so each end is a square with side length 55 cm.

1. The problem asks us to find the length of side $YZ$ in a right triangle $XYZ$ where the right angle is at vertex $Y$. 2. According to Pythagoras' theorem, in a right triangle, t

1. The problem states that two angles in a triangle are 50 degrees and 80 degrees, and we need to find the ratio of the third angle to the sum of the first two angles. 2. Recall th

1. **Problem statement:** (a) Given lines AB and CD intersect at O, with \(\angle AOC = 68^\circ\), find \(\angle BOD\) and explain why.

1. **Problem Statement:** Given quadrilateral ABCD with diagonal AC, points E and F lie on AC such that E is closer to D and F is between A and E. We know lengths: $BD=12$, $AE=8$,

1. **Problem:** Find the volume of a square pyramid with base edge 6 inches and height 5 inches. 2. **Formula:** The volume $V$ of a square pyramid is given by