📐 geometry

1. **Problem statement:** We have five vertices of a hexagon plotted on a coordinate plane: $A(1,5)$, $B(4,1)$, $C(9,1)$, $D(12,9)$, and $E(4,9)$. We need to find the coordinates o

1. **Problem statement:** We have five vertices of a hexagon plotted on a coordinate grid: (1,5), (4,1), (4,9), (9,1), and (9,9). We need to find the coordinates of the sixth verte

1. **State the problem:** Find the equations of the tangent lines $l_1$ and $l_2$ to the circle given by

1. **State the problem:** We need to find the total surface area of a cuboid with length $6$ cm, width $4$ cm, and height $5$ cm. 2. **Formula for total surface area of a cuboid:**

1. **State the problem:** We need to find the volume of a cylinder with radius $r = 9$ cm and height $h = 20$ cm. 2. **Formula for the volume of a cylinder:**

1. **State the problem:** Find the volume of a triangular prism with a triangular base of base length 7 cm and height 6 cm, and prism length 10 cm. 2. **Formula:** The volume $V$ o

1. **Problem Statement:** We are given a circle with center $O$ and points $P$, $Q$, and $R$ on the circle. The angles at the center $O$ subtended by chords $PR$ and $PQ$ are $60^\

1. **Problem:** If the diagonals of a quadrilateral bisect each other at right angles, then name the quadrilateral. 2. **Formula and rules:** A quadrilateral whose diagonals bisect

1. **Problem:** If the diagonals of a quadrilateral bisect each other at right angles, then name the quadrilateral. 2. **Formula and Rules:**

1. **State the problem:** We need to find the bearing of point D from point C using the protractor diagram. 2. **Understanding bearings:** Bearings are measured clockwise from the

1. **Problem statement:** We need to find the bearing of point A from point B, and the bearing of point B from point A.

1. State the problem: We need to arrange the steps for drawing a bearing of 075° from a point X in the correct order. 2. Understand the concept: A bearing is measured clockwise fro

1. **Problem statement:** In triangle $ABC$, point $D$ lies on $AC$ such that $BD \perp AC$ and $\angle DBC = 77^\circ$. Point $E$ lies on $BC$ such that $\angle CAE = 54^\circ$. W

1. **Problem Statement:** We are given triangle MTR with point T above segment MR. The triangle has two equal sides: MT = TR, and angles at T are 35° and 25° on either side. Points

1. **State the problem:** We need to find the measure of angle $\angle MNT$ in the given triangle configuration. 2. **Analyze the diagram and given information:**

1. **State the problem:** Given that $m\angle BHG = 100^\circ$, find $m\angle CED$ in the given quadrilateral with intersecting diagonals and marked congruent segments and right an

1. **State the problem:** Find the angle between the x-axis and the line joining the points $(3, -1)$ and $(4, -2)$. 2. **Formula used:** The angle $\theta$ between the x-axis and

1. **Problem:** A cube has an edge length of 2 cm. Find the total surface area of the cube. 2. **Formula:** The total surface area $S$ of a cube with edge length $a$ is given by:

1. **State the problem:** We have a circular garden with radius $1.5$ m and a regular hexagonal patio inscribed inside it. We need to find the area covered by grass, which is the a

1. **Problem statement:** We have a square with a triangle inside it formed by a diagonal. The triangle has angles 45° and 70°, and we need to find the unknown angle $x$ inside the

1. **State the problem:** We need to show that the points (4, 4), (3, 5), and (-1, -1) form a right-angled triangle without using the Pythagoras theorem. 2. **Approach:** Instead o