If isosceles angles are 45, 67.5 and 67.5 with a height of 5.25" what is the length of the base?
We can take advantage of the symmetry of the isosceles triangle.
By cutting the isosceles triangle in two congruent triangles, each right triangle has a long leg measuring 5.25", (height, H).
The short leg is half the base, K.
Using trigonometry and the definition of tangent, tan(45/2°)=K/H.
=>
K=H*tan(22.5°)=2.187
The base is therefore, 2K, or twice 2.187.
To find the length of the base, we can use the formula for the area of an isosceles triangle, which is given by:
Area = (1/2) * base * height
Since we know the height is 5.25 inches, and the angles are 45°, 67.5°, and 67.5°, we can draw a diagram of the triangle to see that the base is divided into two equal lengths by the perpendicular bisector from the top vertex (where the 45° angle is). Let's call each of those equal lengths "x".
First, we need to find the length of the third side of the triangle. We can use the Law of Cosines, which states that for a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2 * a * b * cos(C)
Since we have an isosceles triangle, with two sides of length x (which we haven't determined yet) and an angle of 67.5° opposite the third side, we can solve for x:
x^2 = x^2 + x^2 - 2 * x * x * cos(67.5)
Simplifying,
x^2 = 2x^2 - 2x^2 * cos(67.5)
Dividing both sides by x^2,
1 = 2 - 2 * cos(67.5)
Rearranging,
cos(67.5) = 2 - 1
cos(67.5) = 1
Since the cosine of 67.5° is not 1, there seems to be a mistake in the given information. The angles and the height provided are not consistent with the dimensions of an isosceles triangle. Please double-check the given measurements.
To find the length of the base of an isosceles triangle, you can use trigonometry.
In this case, we know that the two equal angles are 67.5 degrees. Let's call the length of the base "b". Since the triangle is isosceles, the two sides adjacent to the equal angles are equal in length, so we can call them "a". The height of the triangle is given as 5.25 inches.
Now, we can use the trigonometric function tangent (tan) to find the length of the base. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.
In this triangle, the height is the opposite side to the angle of 67.5 degrees, and the base is the adjacent side. Therefore, we can write:
tan(67.5 degrees) = 5.25 inches / b
To find "b", we can rearrange the equation:
b = 5.25 inches / tan(67.5 degrees)
Using a scientific calculator or an online calculator, we can calculate the value of "b".
Note: Make sure your calculator is set to degrees mode before using the tangent function.
Plugging in the values:
b = 5.25 inches / tan(67.5 degrees)
≈ 5.25 inches / 2.44
Calculating this expression gives us:
b ≈ 2.15 inches
Therefore, the length of the base of the isosceles triangle is approximately 2.15 inches.