An isosceles triangle has legs 4 feet long. If the vertex angle measures 30 degrees, find the length of the base.
let the base be 2x
Construct an altiude to the base, resulting in a right-angled triangle with hypotenuse 4, angle 15° and opposite side x
sin15 = x/4
x = 4sin15
so the base is 2x, or 8sin15°
you do the button pushing.
To find the length of the base of an isosceles triangle, we need to use the properties of this type of triangle. Since the triangle is isosceles, it has two equal legs.
Given that the length of each leg is 4 feet, we can draw the triangle and label it as follows:
|\
| \
4 | \ 4
| \
|____\
30°
To find the length of the base, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposing angle is constant. In this case, we can use the following formula:
(sin(angle) / length of leg) = (sin(opposite angle) / length of opposite leg)
Let's substitute the values we know into the formula:
(sin(30°) / 4) = (sin(30°) / x)
We can simplify the equation by cross multiplying:
4 * sin(30°) = x * sin(30°)
Using the property of sin(30°) = 1/2, we can further simplify the equation:
4 * (1/2) = x * (1/2)
2 = x * (1/2)
Now, we can solve for x by multiplying both sides of the equation by 2:
2 * 2 = x
4 = x
Therefore, the length of the base of the isosceles triangle is 4 feet.
To find the length of the base of the isosceles triangle, we can use the trigonometric relationship between the sides and angles in a right triangle. Here's how you can do it:
1. Draw a triangle with two legs that are 4 feet long, meeting at a point. Label the legs as A and B, and the vertex angle as C.
2. Since this is an isosceles triangle, the two base angles are equal. Therefore, each base angle measures (180 - 30) / 2 = 150 / 2 = 75 degrees.
3. Now, we have a right triangle with one leg measuring 4 feet and an acute angle of 75 degrees.
4. We can use the trigonometric function tangent (tan) to find the length of the base. The tangent of an angle is the ratio of the length of the opposite side to the adjacent side.
tan(75 degrees) = length of the opposite side (which is the length of the base) / length of the adjacent side (which is 4 feet)
5. Rearrange the formula to isolate the length of the base. Multiply both sides of the equation by 4 feet:
length of the base = 4 feet * tan(75 degrees)
6. Use a calculator to find the value of tan(75 degrees):
tan(75 degrees) ≈ 3.73205
7. Multiply the result by 4 feet to find the length of the base:
length of the base ≈ 3.73205 * 4 feet ≈ 14.93 feet
Therefore, the length of the base of the isosceles triangle is approximately 14.93 feet.