📐 geometry

1. **State the problem:** We are given a triangle ABC with AB = CB, angle ABC = 50°, and the bearing of B from A is 070°. We need to find the bearing of C from A. 2. **Understand t

1. **State the problem:** We have a laser beam inside an optical fibre with parallel sides. The beam bounces off the sides at the same angle it hits them. Given the angle of incide

1. **State the problem:** We need to find the volume of a prism with a hexagonal base. 2. **Given data:**

1. **State the problem:** We need to find the volume of a prism given its cross-sectional area and length. 2. **Formula:** The volume $V$ of a prism is calculated by multiplying th

1. **State the problem:** We need to find the size of angle $q$ in a composite figure consisting of a triangle and an irregular pentagon sharing a side. 2. **Identify known angles:

1. The problem asks to find the size of angle $d$ in a regular octagon and to explain Sonia's mistake. 2. Sonia calculated $d$ as $\frac{360}{8} = 45^\circ$. This is correct for th

1. **Problem statement:** Calculate the perimeter of a right-angled triangle with legs of lengths 7 m and 16 m. 2. **Formula:** The perimeter $P$ of a triangle is the sum of the le

1. **State the problem:** Calculate the surface area of a triangular prism with given dimensions: height of the triangle = 8 cm, base width = 6 cm, side width = 9 cm, and length of

1. The problem is to find the area of a right-angled triangle with legs measuring 9 m and 16 m. 2. The formula for the area of a right-angled triangle is:

1. **State the problem:** We are given four figures: a triangular pyramid (tetrahedron), a rectangular prism, a cube, and another rectangular prism. We need to determine which stat

1. **Problem Statement:** We analyze two robotic tracks (lines L1 and L2) and two safety barriers (planes \(\Pi_1\) and \(\Pi_2\)) to determine their spatial relationships and dist

1. **Problem Statement:** We need to find the equation of a plane passing through point $P(2, -1, 4)$ and perpendicular to vector $\vec{n} = \langle 3, 2, -1 \rangle$.

1. **State the problem:** We need to find the distance between diagonally opposite corners of a rectangular brick wall that is 6 m wide and 1 m tall. 2. **Formula used:** We use Py

1. **Problem statement:** We have a right triangle XYZ with a right angle at Y. The sides XY and XZ are given as 9 cm and 16 cm respectively, and we need to find the length of side

1. **State the problem:** We need to find the size of angle $\alpha$ and then use it to find the size of the reflex angle $b$.

1. The problem asks to find the value of $x$ in the circle with center $O$ and points $A, B, C, D$ on the circumference. 2. We know that the angle at the center $O$ subtended by ch

1. **State the problem:** We have a right triangle PQR with a right angle at R. The hypotenuse PQ measures 37 cm, and one leg RQ measures 35 cm. We need to find the length of the o

1. **Problem statement:** Given points $P(x_1,y_1)$, $Q(x_2,y_2)$, and $R(x,y)$ lie on the circumference of a circle with $PQ$ as the diameter, show the equation of the circle. 2.

1. **State the problem:** We need to find the area of a circle with a diameter of 16 mm. 2. **Recall the formula for the area of a circle:**

1. **State the problem:** We need to find the area of a circle with a diameter of 14 mm. 2. **Recall the formula for the area of a circle:**

1. The problem states that a metal cube has a volume of 96 cm^3, correct to the nearest cm^3. We need to find the least possible length of each side of the cube, $x$, correct to 2