📐 geometry

1. The problem asks if lines $j$ and $k$ are parallel and to explain the reasoning. 2. From the graph description, lines $j$ and $k$ are marked as parallel ($j \parallel k$).

1. The problem involves finding the values of angles $h$, $g$, and $f$ formed by two intersecting lines with given angles $100^\circ$ and $33^\circ$. 2. Vertically opposite angles

1. **State the problem:** We have triangle \(\triangle ABC\) with sides \(AB = x\) cm, \(BC = x + 2\) cm, \(AC = 5\) cm, and angle \(\angle ABC = 60^\circ\). We need to find \(x\).

1. **State the problem:** We are given that point P(4,2) is the midpoint of the line segment OPC, where O is the origin (0,0). We need to find the coordinates of point C. 2. **Reca

1. **State the problem:** We need to find the length of side $AB$ in triangle $ABC$ where $AC=6.5$ cm, $BC=8.7$ cm, and the angle $ACB=100^\circ$. 2. **Identify the formula:** Use

1. The problem asks to find the area of a circle with radius 7 cm, using \(\pi = \frac{22}{7}\).\n\n2. The formula for the area of a circle is \(A = \pi r^2\), where \(r\) is the r

1. **State the problem:** We need to find the angle between the hour and minute hands of a clock at 6:30. 2. **Calculate the minute hand angle:** The minute hand moves 360 degrees

1. **Problem Statement:** Two circles intersect at points C and D. ABCD is a cyclic quadrilateral in the first circle. The line AD extended meets the second circle at E, and the li

1. **Problem statement:** Two circles intersect at points C and D. Quadrilateral ABCD is cyclic in the first circle. The line AD extended meets the second circle at E, and the line

1. **State the problem:** We need to find the length of side AB in triangle ABC where AC = 6.5 cm, BC = 8.7 cm, and angle ACB = 100 degrees. 2. **Identify the known values:**

1. **Problem:** Find the area of isosceles triangle $\triangle PQR$ where $PE \perp QR$, $PE=1.6$ cm, and $QR=5.5$ cm. Step 1: The area of a triangle is given by $$\text{Area} = \f

1. State the problem: Find the area of a rectangle with length 15 m and width 12 m. 2. Recall the formula for the area of a rectangle: $$\text{Area} = \text{length} \times \text{wi

1. **Problem:** Find the area of isosceles triangle ΔPQR where PE ⊥ QR, PE = 1.6 cm, QR = 5.5 cm. Step 1: Since PE is perpendicular to QR, PE is the height of the triangle.

1. **State the problem:** Calculate the surface area (excluding the top rectangle) and volume of a half cylinder with radius $r=34$ cm and length $l=0.58$ m. Convert all units to c

1. **State the problem:** We have a triangle with points B, C, G where \(\triangle BCG\) is isosceles with \(|BC| = |CG|\) and \(\angle BCG = 36^\circ\). Line segment \(BD\) is par

1. The problem states that we have two similar triangles with areas 140.8 cm² and 178.2 cm², and we need to find the ratio of their corresponding medians. 2. For similar triangles,

1. **State the problem:** We are given a diagram with angles at point Q. We know \(\angle PQR = 90^\circ\), \(\angle PQR = 32^\circ\) (likely a typo, assume \(\angle PQS = 32^\circ

1. **State the problem:** We have two similar triangles with corresponding angle bisectors of lengths 3 and 5. The area of the smaller triangle is 45. We need to find the area of t

1. The problem is to show the ratio of the sides of a triangle or geometric figure. 2. Typically, side ratios are expressed as fractions or decimals comparing the lengths of sides.

1. **State the problem:** We have two chords AC and BD intersecting at point O inside a circle. We need to find the ratio of the areas of triangles AOB and COD. 2. **Recall propert

1. **Problem statement:** We have three similar triangles with side lengths in the ratio $3:4:5$. The sum of their areas is 300 cm². We need to find the area of the smallest triang