Length of AB = √[(4-1)² + (6-2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Using the cosine rule: c^2 = a^2 + b^2 - 2ab*cos(C) = 10^2 + 24^2 - 2*10*24*(1/2) = 100 + 576 - 240 = 436. Thus, c = √436 = 20.
Perimeter = a + b + c = 5 + 12 + 13 = 30.
Since 7² + 24² = 49 + 576 = 625 = 25², triangle ABC is a right triangle.
Using Heron's formula, s = (7 + 24 + 25)/2 = 28. Area = √[s(s-a)(s-b)(s-c)] = √[28(28-7)(28-24)(28-25)] = √[28*21*4*3] = 84.
Since 8² + 15² = 17², triangle ABC is a right triangle.
Since 8² + 15² = 64 + 225 = 289 = 17², triangle ABC is a right triangle.
Since 10² + 24² = 100 + 576 = 676 = 26², triangle ABC is a right triangle.
Perimeter = 8 + 15 + 17 = 40 cm.
Using Heron's formula, the semi-perimeter s = (8 + 15 + 17)/2 = 20. Area = √[s(s-a)(s-b)(s-c)] = √[20(20-8)(20-15)(20-17)] = √[20*12*5*3] = 60.
The perimeter of a triangle is the sum of its sides. Therefore, perimeter = a + b + c = 5 + 12 + 13 = 30.
The perimeter of a triangle is the sum of its sides: a + b + c = 8 + 15 + 17 = 40.
Using the cosine rule, cos A = (b² + c² - a²) / (2bc) = (15² + 17² - 8²) / (2 * 15 * 17) = 0.8.
Since the sides are in the ratio of a Pythagorean triplet (3, 4, 5), triangle ABC is a right triangle.
The sides satisfy the Pythagorean theorem, hence it is a right triangle.
A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.
Angle O = 180 - (angle M + angle N) = 180 - (45 + 45) = 90 degrees.
Using Heron's formula, s = (10 + 24 + 26)/2 = 30. Area = √(30(30-10)(30-24)(30-26)) = √(30*20*6*4) = 120 cm².
Since 8^2 + 15^2 = 17^2, triangle XYZ is a right triangle.
Area = 1/2 * base * height => 24 = 1/2 * 8 * height => height = 6 cm.
Area = 1/2 * base * height => 30 = 1/2 * 10 * height => height = 6 units.
Since 7^2 + 24^2 = 25^2, triangle ABC is a right triangle.
The area of an equilateral triangle is given by the formula (√3/4)a².
For a right triangle, the circumradius R = hypotenuse/2 = 13/2 = 6.5.
Circumradius R = (abc)/(4K), where K is the area. Area K = 24 (using Heron's formula). R = (6*8*10)/(4*24) = 5.
The circumradius R of a triangle can be calculated using the formula R = (abc)/(4 * Area). Here, Area = 84 cm², so R = (7 * 24 * 25)/(4 * 84) = 13.
The circumradius R of an equilateral triangle is given by R = a/(√3).
