The area of a right triangle is 50. One of its angles is 45 degree. Find the lengths of the sides and hypotenuse of the triangle.

The triangle is right and the size one of its angles is 45 degree; the third angle has a size 45 degree and therefore the triangle is right and isosceles. Let x be the length of one of the sides and H be the length of the hypotenuse. Area = (1/2)x^2 = 50 , solve for x: x = 10 We now use Pythagoras to find H: x^2 + x^2 = H^2 Solve for H: H = 10 sqrt(2)

What are the values of the real numbers a, b and c if the equation - 4 x(x + 5) - 3(4x + 2) = a x^2 + b x + c is true for all values of x?

Expand the left side of the given equation and group like terms. - 4x^2 - 20x - 12 x - 6 = a x^2 + b x + c - 4x^2 - 32x - 6 = a x^2 + b x + c The two polynomials on each side of the equation are equal if the corresponding coefficients (of the same power of x) are equal. Hence Answer: a = - 4 , b = - 32 and c = - 6

Find the slope of the line passing through the points (-1, -1) and (2, 2).

Given the points (-1, -1) and (2, 2), the slope m is given by m = (y2 - y1) / (x2 - x1) = (2 - -1) / (2 - -1) = 1