An isosceles triangle has a vertical angle of 116 degrees and its base is 8cm long .calculate its height
Just draw a triangle
Lebel it ABC (how ever you want to lebel it)
If A=116
Then if you divide the triangle into two parts
The base is now 4 for the first part
And 4 for the second part
A/2=58
Now you have a two set of triangle
Tan58=4/h
h=4/tan58=?
To calculate the height of an isosceles triangle, we can use the properties of a right-angled triangle formed by the height, half of the base, and the hypotenuse.
Given that the base of the isosceles triangle is 8 cm and the vertical angle is 116 degrees, let's calculate the height using trigonometry.
Step 1: Find the measure of the base angles of the isosceles triangle.
Since the isosceles triangle has two equal angles, we can find the measure of each base angle by subtracting the vertical angle from 180 degrees, and then dividing by 2.
Base Angle = (180 degrees - Vertical Angle)/2 = (180 - 116)/2 = 64/2 = 32 degrees
Step 2: Find the length of one of the legs of the right-angled triangle formed by the height and half of the base.
Using the sine function, we can find the length of the leg:
sin(32 degrees) = opposite/hypotenuse
opposite = sin(32 degrees) * hypotenuse
The hypotenuse is half of the base length, so:
hypotenuse = 8 cm / 2 = 4 cm
Substituting the values, we get:
opposite = sin(32 degrees) * 4 cm
Step 3: Calculate the height of the isosceles triangle.
Using the Pythagorean theorem, we can find the height (which is the other leg of the right-angled triangle):
height^2 = hypotenuse^2 - opposite^2
height^2 = (4 cm)^2 - (sin(32 degrees) * 4 cm)^2
Calculating:
height^2 = 16 cm^2 - (sin(32 degrees))^2 * 16 cm^2
height^2 = 16 cm^2 - (0.5299... * 16 cm)^2
height^2 ≈ 16 cm^2 - (8.479... cm)^2
height^2 ≈ 16 cm^2 - 71.91 cm^2
height^2 ≈ -55.91 cm^2
Since we cannot have a negative value for the height, it seems that there may be an error in the input or calculation. Please double-check the given information and perform the calculation again.
To calculate the height of an isosceles triangle, we can use the formula for the area of a triangle. The formula is:
Area = (base * height) / 2
In this case, we know the base of the triangle is 8 cm. We need to find the height.
To find the height, we need to use trigonometry. Since we know that the triangle is isosceles, we can conclude that the two remaining angles are equal. Let's call them angle A and angle B.
Since the vertical angle is 116 degrees, angle A and angle B are equal to half of that, which is 116 / 2 = 58 degrees.
Now, we can use trigonometry to find the height of the triangle.
Let's consider triangle ABD, where A is the top vertex of the triangle, B and D are the endpoints of the base, and AD is the height.
In triangle ABD, we have:
Angle ABD = 90 degrees
Angle BAD = 58 degrees
We can use the trigonometric function tangent (tan) to find the height AD of the triangle:
tan(BAD) = height / base
Substituting in the known values:
tan(58) = AD / 8
To solve for AD, we multiply both sides by 8:
AD = 8 * tan(58)
Now, we can calculate the value of AD using a calculator or by using a trigonometric table.
Therefore, the height of the isosceles triangle is AD = 8 * tan(58) cm.