The vertex angle of an isosceles triangle is 40°20'. Each of the leg measures 320 mm. Find the length of the base.
each half of the base is 320 sin 20°10'
A = B = (180-40.33)/2 = 69.83o = Base angles.
sinA/a = sinC/c
sin69.83/320 = sin40.33/c
c*sin69.83 = 320*sin40.33
c = 320*sin40.33/sin69.83 = 220.6 mm
Or
cosA = 0.5*AB/AC
AC*cosA = 0.5*AB
AB = 2AC*cosA = 2*320*cos669.83=220.7 mm
To find the length of the base of an isosceles triangle, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides of lengths a, b, and c, and opposite angles A, B, and C respectively, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, the isosceles triangle has two equal legs of length 320 mm, and the vertex angle is 40°20'. To find the length of the base, we need to know the length of the other two sides. Since the triangle is isosceles, we can divide it into two congruent right triangles by drawing an altitude from the vertex angle to the base.
Let's assume the length of the base is x. All sides in each of the congruent right triangles would have a hypotenuse of x, and one of the other sides would be 320 mm.
Using trigonometry, we can find the length of the other side of each right triangle.
In a right triangle, the cosine of an angle is defined as the adjacent side divided by the hypotenuse. In our case, the adjacent side is 320 mm and the hypotenuse is x. Therefore, we have:
cos(40°20') = 320 / x
To find x, we can rearrange the equation:
x = 320 / cos(40°20')
Using a calculator, we can find the value of cos(40°20'):
cos(40°20') ≈ 0.7590
Substituting this value into the equation, we get:
x = 320 / 0.7590 ≈ 421.15 mm
Therefore, the length of the base of the isosceles triangle is approximately 421.15 mm.
To find the length of the base of an isosceles triangle, we can use the Law of Cosines.
The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, the isosceles triangle has legs of equal length, so we can label each leg as "a" and the base as "b." Let's solve for the length of the base, which is "b."
Given information:
- Vertex angle = 40°20'
- Measure of each leg = 320 mm
First, let's convert the angle from degrees and minutes to decimal degrees.
0.20 minutes is equal to 1/60 of a degree: 0.20/60 = 0.00333 degrees
Adding this to 40°, we get: 40 + 0.00333 = 40.00333°
Now, we can substitute the values into the Law of Cosines equation:
b^2 = a^2 + a^2 - 2*a*a * cos(C)
Since the triangle is isosceles, both legs are of equal length:
b^2 = 320^2 + 320^2 - 2*320*320 * cos(40.00333°)
Calculating the values:
b^2 = 102400 + 102400 - 2*320*320 * cos(40.00333°)
b^2 = 204800 - 204800 * cos(40.00333°)
Now, we need to calculate the value of cos(40.00333°). You can use a scientific calculator or an online calculator to find the cosine of the angle.
Once you have the value of cos(40.00333°), plug it back into the equation:
b^2 = 204800 - 204800 * <cosine value>
Finally, take the square root of b^2 to find the length of the base, "b".