2.What is the value of x? Enter your answer, as a decimal, in the box. 3.What is the value of x? 4.What is the value of x? 5.The area of a right triangle is 270 m². The height of the right triangle is 15 m. What is the length of the hypotenuse of the right triangle?

I am going to assign the first image to problem 2, the second image to problem 3, and the third image to problem 4 and solve for x in each image. Basically, all of these problems can be solved used the law of sines which is as follows:

a/sinA = b/sinB = c/sinC

This states that the length of a side, divided by the sine of the opposite angle, is the same for every side in a triangle.

2.) We must solve for side x in the smaller triangle. We know two sides of the large triangle, and one side of the smaller triangle. The two parallel lines tell us that angle A = angle N and angle B = angle P. The side of the smaller triangle that we do know is, 67.2 - 32 = 35.2 m. Now we can plug values into the law of sines.

35.2/sinB = x/sinA
sinA = (x/35.2)sinB

81.9/sinB = 67.2/sinA
sinA = (67.2/81.9)sinB

We can now equate both sinA equations and solve for x.

(x/35.2)sinB = (67.2/81.9)sinB
x/35.2 = 67.2/81.9
x = (35.2)(67.2/81.9)
x = 42.9 m

3.) We will make the assumption that the angle divided by the line inside the triangle is split in equal halves, therefore, both angles are the same. The angles at point D are angle D and and 180-D. It turns out that sinD = sin(180-D). If you do not believe this principle, test it out in a calculator. This will simplify our problems. We can simply use the law of sines once more.

(x+4)/sinE = 44.8/sinD
sinE = ((x+4)/44.8)sinD

35/sinE = 56/sinD
sinE = (35/56)sinD

((x+4)/44.8)sinD = (35/56)sinD
(x+4)/44.8 = 35/56
x+4 = (35/56)(44.8)
x = (35/56)(44.8) - 4
x = 24 m

4.) This problem is very similar to problem two, containing a parallel line that results in angles on the same side of the parallel line being equivalent.

45/sinθ = (13+2x)/sinθₓ
sinθ = (45/(13+2x))sinθₓ

5/sinθ = 3/sinθₓ
sinθ = (5/3)sinθₓ

(5/3)sinθₓ = ((45/(13+2x))sinθₓ
5/3 = 45/(13+2x)
13+2x = 27
2x = 14
x = 7m

5.) The area of a triangle is found using the formula, A = (1/2)b·h. We are already given the area of the triangle and the height of the triangle. The base of this triangle IS the hypotenuse, so solving for the base will answer this question.

A = (1/2)(b)(15) = 270
b = (270)(2/15)
b = 36 m