1. To find the length of each congruent side of the isosceles triangle, we can use the properties of isosceles triangles.
Since the triangle is isosceles, it has two congruent sides. Let's assume the length of each congruent side is 'x'.
We know that the angle opposite the base measures 86 degrees. In an isosceles triangle, the angle opposite the base is also congruent to each other, so the other two angles will also measure 86 degrees each.
Now, let's use the Law of Cosines to solve for 'x'. The Law of Cosines states that in a triangle with sides a, b, and c and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In our case, the two congruent sides have length 'x'. The base has a length of 14 cm. The angle opposite the base is 86 degrees.
Plugging in the values into the Law of Cosines equation:
x^2 = 14^2 + 14^2 - 2*14*14*cos(86)
Simplifying:
x^2 = 196 + 196 - 392*cos(86)
x^2 = 392 - 392*cos(86)
Now, we need to find the value of cos(86). Using a calculator, find the cosine of 86 degrees:
cos(86) ≈ 0.08716
Now substitute this value back into the equation:
x^2 = 392 - 392 * 0.08716
x^2 = 392 - 34.19232
x^2 ≈ 357.80768
Taking the square root of both sides, we get:
x ≈ √357.80768
x ≈ 18.92
So, the length of each congruent side of the isosceles triangle is approximately 18.92 cm.
2. To find the vertical drop on the ski slope, we can use trigonometry and the given information of the incline and the length of the slope.
The incline angle of the ski slope is 42 degrees, and the length of the slope is 170 yards.
The vertical drop can be found using the following trigonometric relationship:
Vertical Drop = Length of Slope * sin(Incline Angle)
Plugging in the values:
Vertical Drop = 170 * sin(42)
Now, find the sine of 42 degrees using a calculator:
sin(42) ≈ 0.6691306
Vertical Drop = 170 * 0.6691306
Vertical Drop ≈ 113.5531212
Therefore, the vertical drop on the ski slope is approximately 113.55 yards.