A right triangle with coordinates A (－1, 4), B(－1, 8) and C (3, 4) is first rotated 90 degrees counterclockwise and then translated 2 units left and 3 units down to form Triangle A’B’C’. What is the measure of angle B’A’C’, in degrees, in the resulting figure?

The ∠B prime A prime C prime  in the resulting figure = 90°

Step-by-step explanation:

The coordinates of the right triangle = A( -1, 4), B(-1, 8) and C(3, 4)

The rotation 90° counterclockwise of a point on a preimage (x, y) gives the coordinates of the location of the point of the image after rotation as (-y, x)

Therefore we have;

Applying 90° counterclockwise rotation we have;

Applying a translation T(-2, -3) to the new points above which is a translation 2 units left and 3 units down, we have;

The coordinates of the point A prime is (-4 - 2, -1 - 3) = (-6, -4)

The coordinates of the point B prime is (-8 - 2, -1 - 3) = (-10, -4)

The coordinates of the point C prime is (-4 - 2, 3 - 3) = (-6, 0)

The coordinates of the vertices of the triangle ΔA prime B prime C prime are A prime (-6, -4) B prime (-10, -4) C prime (-6, 0)

The measure of the angle ∠B prime A prime C prime  is given as follows;

Length of a segment of each segment of the triangle are found using the following equation;

Which gives;

Length of B prime A prime = √(((-10) - (-6))² + ((-4) - (-4))²) = 4

Length of B prime C prime = √(((-10) - (-6))² + ((-4) - 0)²) = 32 = 4·√2

Length of C prime A prime = √(((-6) - (-6))² + ((0) - (-4))²) = 4

∴ B prime C prime is the hypotenuse side and the ∠B prime A prime C prime = The angle opposite to the hypotenuse side = 90°

The ∠B prime A prime C prime  in the resulting figure = 90°

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).