1. The base of an isosceles triangle is 14 cm in length and the angle opposite the base measures 86 degrees. Find the length of each of the congruent sides.
2. A ski slope has a 42 degrees incline and 170 yard long. Find the vertical drop.
draw in an altitude to the base , which would bisect the base.
So now you have a right angled triangle,
let the equal sides be x cm each
sin 43° = 7/x
x = 7/sin43 = appr 10.26 cm
For the second, I will assume that the 170 yrds is measured along the ski run
then
sin42 = x/170
x = 170sin42 = appr 113.75 yrds
1. Well, if the angle opposite the base measures 86 degrees, then the other two angles must be a little jealous! They both must measure (180 - 86) / 2 = 47 degrees. Now, to find the length of each congruent side, we need to use a little trigonometry. Since we have the base length and an angle, we can use the sine function. So, sin(47) = length of congruent side / 14cm. Cross-multiplying, we get length of congruent side = sin(47) * 14cm. Plug in your calculator and voila! You've got your answer.
2. Ah, the ski slope. Let's calculate the vertical drop and slide into the answer! We know the length of the slope (170 yards) and the angle of incline (42 degrees). To find the vertical drop, we can use a bit of trigonometry again. Remember, the vertical drop is the opposite side to the angle of incline. So, we use the sine function: sin(42) = vertical drop / 170 yards. Cross-multiplying, we get vertical drop = sin(42) * 170 yards. Calculate that, and you'll know how far your heart will drop on that slope!
1. To find the length of each of the congruent sides of an isosceles triangle, we can use the angle opposite the base.
Let's label the congruent sides as "x" and the base as "y".
In an isosceles triangle, the two congruent sides are equal in length.
Using the properties of isosceles triangles, we can find the length of the congruent sides using the law of sines.
The law of sines states that in any triangle:
(sin A) / a = (sin B) / b = (sin C) / c
In our case, we know that the angle opposite the base is 86 degrees, and the base is labeled as "y".
So, we have:
(sin 86) / x = (sin 47) / y = (sin 47) / 14
To find the length of the congruent sides, we can use the equations above.
2. To find the vertical drop of the ski slope, we can use trigonometric functions.
The incline of the slope is given as 42 degrees, and the length of the slope is 170 yards.
We can use the sine function to find the vertical drop:
sin(incline) = vertical drop / length of slope
sin(42) = vertical drop / 170
To find the vertical drop, we can rearrange the equation and solve for it:
vertical drop = sin(42) * 170
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.