📐 geometry

1. The problem involves verifying the angle values in a diamond shape inscribed in a circle, with given angles 48°, 48°, 96°, and 84°. 2. First, check the sum of angles in the diam

1. **Problem (a):** Find $x$ and $y$ in the triangle with angles $44^\circ$, $y$, and an exterior angle $68^\circ$ adjacent to $x$. 2. The exterior angle $68^\circ$ equals the sum

1. **State the problem:** We have two similar tanks with capacities 1,000,000 litres and 512,000 litres respectively. The smaller tank has a surface area of 1200 m². We need to fin

1. **Problem statement:** Given cyclic quadrilateral ABCD with center O, AB and CD extended meet at E, DC=BC, and \(\angle BCE=48^\circ\). Find angles \(\angle BAD\), \(\angle BDC\

1. **State the problem:** We have a regular pentagon with an inscribed circle (incircle) of radius $r=5$ cm. We need to find the area of the pentagon outside the circle, i.e., $\te

1. **State the problem:** We have a sector of a circle with area 550 cm². This sector is curved to form an open cone with radius 7 cm. We need to find the height of the cone. 2. **

1. **State the problem:** We need to find the perpendicular bisectors of the lines $x+y=0$ and $x-y=0$, which intersect at the origin, and analyze the right triangle with vertices

1. **State the problem:** We are given two lines $x+y=0$ and $x-y=0$ intersecting at the origin, which is the circumcenter of a right triangle with vertex $A(5,7)$. 2. **Identify t

1. সমস্যাটি হলো: একটি রম্বসের দুইটি কর্ণ ৭ সেমি এবং ৪ সেমি, এবং একটি কোণ ৬০ ডিগ্রী দেওয়া আছে। রম্বসটির ক্ষেত্রফল নির্ণয় করতে হবে। 2. রম্বসের ক্ষেত্রফল নির্ণয়ের সূত্র হলো: $$\tex

1. The problem is to find the area of a triangle with a base of 6 cm and a height of 4.5 cm. 2. Recall the formula for the area of a triangle: $$\text{Area} = \frac{1}{2} \times \t

1. **State the problem:** We are given two intersecting lines forming opposite angles. One angle measures 74° and the opposite angle measures $(3x + 4)°$. We need to find the value

1. **State the problem:** We have two intersecting lines forming vertical angles. One angle measures 74° and the opposite angle measures $x + 4^\circ$. We need to find the value of

1. **State the problem:** We need to find the length of the diagonal of a rectangle with length $6$ units and width $8$ units. 2. **Recall the formula:** The diagonal $d$ of a rect

1. **Problem Statement:** We have a pentagon ABCDE with vertices \( A(-3, -1), B(-3, 5), C(1, 8), D(5, 5), E(5, -1) \). We need to find its area and perimeter, then dilate it by a

1. **State the problem:** We are given a point $P(1, -2, 1)$ and a plane defined by the equation $x + 2y - 2z = \alpha$, where $\alpha > 0$. The distance from $P$ to the plane is 5

1. **Problem 1: Construct quadrilateral ABCD where AB \cong CD and AD \cong BC.** Step 1: Draw angle \(\angle DAB\) using a protractor and straightedge.

Problem: Draw a chord in a circle with radius $5$ that subtends a central angle of $60^{\circ}$.\n\n1. Given.\nGiven a circle with center $O$ and radius $r=5$.\nWe want the chord t

1. **State the problem:** We need to construct a rectangle where one diagonal divides the opposite angles into 50 degrees and 40 degrees. 2. **Recall properties of a rectangle:** A

1. The problem is to understand how to draw a chord in a circle. 2. A chord is a straight line segment whose endpoints both lie on the circle.

1. The problem states that PQR is a common tangent to two circles touching externally at point O, and OTU is a straight line. 2. Given angles are \(\angle POS = 50^\circ\) and \(\a

1. **State the problem:** We are given the volume of a cylinder as 402 cubic units and the radius as 4 units. We need to find the height $h$ of the cylinder. 2. **Recall the formul