📐 geometry

1. **State the problem:** We need to find the measure of angle $m\angle DFE$ given a portion of a circle with chord $D$ and points $D$, $F$, and $E$ on or near the circle. 2. **Rec

1. **State the problem:** We need to find the angle $a$ at vertex $C$ in a scalene triangle $ABC$ drawn on a unit grid, where the side lengths are not aligned to the grid. 2. **Ide

1. **State the problem:** We have three similar solid shapes A, B, and C. - Surface area of A = 4 cm²

1. **Find the missing length $x$ in the right triangular prism with volume 432 cm³.** The volume formula for a triangular prism is:

1. **Problem Statement:** Find the missing height $H$ of a right triangular prism with volume $432$ cm³ and base sides $12$ cm and $8$ cm. 2. **Formula:** Volume of a prism is give

1. **Problem Statement:** Reflect an angle over the line $y = -x$ and explain if it is possible. 2. **Reflection Concept:** Reflection over a line in the coordinate plane means eve

1. The problem is to analyze the shape of a chef's hat described as an outline with smooth curved lines forming five rounded lobes at the top and a curved rectangular base. 2. Sinc

1. **State the problem:** We are given two triangles, \(\triangle ABK\) and \(\triangle ACK\), with some side lengths expressed in terms of \(x\). We need to determine if these tri

1. **Problem Statement:** Given that point P is the circumcenter of \(\triangle ABC\), find segments congruent to \(BR\), \(CS\), and \(BP\).

1. **State the problem:** We need to find the hypotenuse $z$ of a right-angled triangle where the two legs are 3 cm and 6 cm. 2. **Formula:** Use Pythagoras' theorem which states:

1. The problem asks to state one condition for two triangles to be similar. 2. One condition for two triangles to be similar is that their corresponding angles are equal (AAA crite

1. **State the problem:** We have a parallelogram with base 10 cm and height 4 cm, cut into three pieces: two right triangles and one rectangle. We want to rearrange these pieces t

1. The problem is to identify the coordinates of points A, B, C, D, and E on the given coordinate plane. 2. The coordinate plane has x and y axes with grid lines at intervals of 4

1. The problem involves a cone with height 12 cm and base radius 2 cm, cut and opened into a sector with angle $x=60^\circ$. We need to find the area of the shaded sector. 2. First

1. **State the problem:** Calculate the area of an irregular polygon with given side lengths in inches. 2. **Approach:** We can find the area by decomposing the polygon into rectan

1. **Stating the problem:** We are given an irregular polygon with the following side lengths: top and bottom horizontal sides are 9 ft each, left vertical side is 6 ft, right vert

1. The problem involves identifying types of transformations or symmetries related to given graphs. 2. The first graph is described as a wavy line, which typically represents a con

1. The problem asks to find the area of each figure described. 2. For the right triangle inside a rectangle with hypotenuse 18 and one angle 30°:

1. The problem involves a rectangle with a diagonal of length 10 units. 2. We want to find the dimensions of the rectangle or verify the relationship between its sides and the diag

1. **Problem statement:** Given quadrilateral PQRS with diagonals PR and QS intersecting at O, and lengths PQ = 4 cm, OQ = 3.5 cm, QR = 7 cm, we need to (i) show that $\cos(\angle

1. **Problem statement:** Given parallelogram PQRS with diagonals PR and QS intersecting at O, where |PQ| = 4 cm, |QR| = 7 cm, and |QO| = 3.5 cm, we need to (i) show that $\cos(\an