📐 geometry

1. Problem 1: Find lengths $a$ and $b$ in triangle ABC where angle $B=45^\circ$, hypotenuse $AC=2$ cm. 2. Since triangle ABC is right angled at $B$, and angle $B = 45^\circ$, the o

1. Stating the problem: We have a hemispherical bowl with an external radius of 18 cm and a thickness of 3 cm. We need to find the volume of the wood making up the bowl. 2. Define

1. **State the problem**: We have a triangle with angles 100°, 38°, and 42°, and one side length of 13 cm opposite the 42° angle. We need to find the length $x$ cm opposite the 38°

1. Stating the problem: We need to calculate three things:

1. The problem states that a trapezium is enlarged by a scale factor of 3 with the center of enlargement at the point marked with a red x. 2. The vertex V is a point on the origina

1. **State the problem:** We need to find the total area of a composite shape made of two rectangles. 2. **Identify the dimensions:**

1. **State the problem:** We have trapezium ABCD with vertices and a centre of enlargement X at (2,8). The trapezium is enlarged by a scale factor of 2 with centre X to get trapezi

1. The problem asks for the size of angle $YXZ$ in a right triangle $YXZ$ where $Z$ is the right angle. 2. Given sides: $YZ = 20$ cm (adjacent to angle $YXZ$), $ZX = 8$ cm (opposit

1. **State the problem:** We need to find the size of angle $YZX$ in triangle $XYZ$, where $\angle X$ is a right angle, $XY = 7.5$ cm, and $ZY = 22.3$ cm. 2. **Identify the sides:*

1. Let's analyze the first triangle with the numbers 5, 8, 9, 9, 6, and 12 on its segments. 2. We can check if these segments satisfy the triangle inequality or if they represent a

1. Given triangle ABC is isosceles with BA = BC. This means angles opposite these sides are equal, so \( \angle BAC = \angle BCA \). 2. Point D lies on AC such that ABD is isoscele

1. **State the problem:** We are given a triangle ABC with lengths AB = $3x - 4$, AC = $2x + 12$, and BC = $7x - 2$. It is given that the ratio $AB : AC = 1 : 2$. We need to show t

1. **State the problem:** We have two right-angled triangles ABC and BCD sharing points B and C. Given:

1. **State the problem:** We have a straight line EGH with points E, G, H such that EG = 20 and GH = 7. Points E, G, H lie on a line, and point F is above G forming two right-angle

1. **State the problem:** The cosine rule (or law of cosines) relates the lengths of sides of a triangle to the cosine of one of its angles. It states: $$c^2 = a^2 + b^2 - 2ab\cos

1. We are given a right triangle XYZ with right angle at O on the base XY. 2. We know:

1. The basic formulas involving angles typically come from geometry and trigonometry. 2. The sum of angles in a triangle is always $$180^\circ$$, so for a triangle with angles $A$,

1. **State the problem:** We have a right-angled triangle XYZ with right angle at Z. Point W lies on XY such that WZ is perpendicular to XY, dividing the triangle into WXZ and WZY.

1. **Stating the problem:** Calculate lengths WY and XY in a right-angled triangle XYZ with right angles at W and Z.

1. **State the problem:** We have two right-angled triangles WZY and WXY sharing vertex W. 2. **Given:** In triangle WZY, legs are $WZ=4$ cm and $ZY=7.5$ cm. Triangle WXY has leg $

1. **State the problem:** Find the minimum distance from the origin $(0,0)$ to the surface given by $$x^2 + y^2 - 1 = 0.$$ This surface represents the unit circle centered at origi