Each base angle of an isosceles triangle measures 42°. The base is 14.6 feet long.
A) Find the length of a leg of the triangle. Round to the nearest tenth of a foot.
B) Find the altitude of the triangle. Round to the nearest tenth of a foot.
A) Well, let's put our clown hats on and solve this puzzle! Each base angle of an isosceles triangle measures 42°, which means the other angle must be 180° - 42° - 42° = 96°.
Now, we can use the Law of Sines to find the length of the leg. Sine of 96° divided by the length of the leg is equal to sine of 42° divided by the length of the base (14.6 feet). Let's do some circus math:
sin(96°) / x = sin(42°) / 14.6
Cross multiplying, we get:
x = (14.6 * sin(96°)) / sin(42°)
Using our trusty clown calculator, we find that x ≈ 23.3 feet (rounded to the nearest tenth).
B) Ah, the altitude! We're going to need some clown aerial tricks for this one. To find the altitude of the triangle, we can split it into two right triangles. The length of the base (14.6 feet) will be the hypotenuse, and we can call the altitude h.
Using some clown Pythagorean magic, we have:
h^2 + (14.6 / 2)^2 = x^2
Simplifying, we get:
h^2 + 7.3^2 = (23.3)^2
Solving for h, we get:
h ≈ √(23.3^2 - 7.3^2)
After some clown calculations, we find that h ≈ 22.6 feet (rounded to the nearest tenth).
So, the length of a leg of the triangle is approximately 23.3 feet, and the altitude is approximately 22.6 feet. Enjoy your geometric circus adventure!
To find the length of a leg of the isosceles triangle, we can use the property of an isosceles triangle where the base angles are equal.
A) To find the length of a leg, we need to use the tangent function. Tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle.
In this case, we have an isosceles triangle, so the two angles opposite the equal sides are both 42°. Let's label one of the legs as x:
tan(42°) = x / (1/2 * 14.6)
First, we divide the base length by 2 since it is only half of the triangle's width (the other half is also equal due to the equal angles).
Now, let's solve the equation for x:
x = tan(42°) * (1/2 * 14.6)
Using a calculator, evaluate the right-hand side of the equation:
x ≈ tan(42°) * 7.3
The approximate value of tan(42°) is 0.9004, so:
x ≈ 0.9004 * 7.3
Calculating the multiplication:
x ≈ 6.52
Therefore, the length of a leg of the isosceles triangle is approximately 6.52 feet (rounded to the nearest tenth).
B) To find the altitude of the triangle, we can use the Pythagorean theorem. The altitude is a perpendicular line from the top vertex of the triangle to the base.
Let's label the altitude as h. We can form a right triangle using one of the legs, the altitude, and half the base length:
Using the Pythagorean theorem:
h² = x² - (1/2 * 14.6)²
Using the previously calculated value of x:
h² = (6.52)² - (1/2 * 14.6)²
Calculating the squares:
h² ≈ 42.5504 - 53.29
Subtracting:
h² ≈ -10.7396
Since the square of a length cannot be negative, there appears to be an error in the problem statement. A triangle cannot exist with the given measurements. Please verify the given information and try again.