In angle ABC,angle A=angle B=32degree and AB=108 mm,find the perpendicular height from C to the baseAB.
if the height is h,
h/54 = tan 32°
To find the perpendicular height from C to the base AB in triangle ABC, we can use trigonometry.
Given:
Angle A = 32 degrees
Angle B = 32 degrees
AB = 108 mm
Since Angle A and Angle B are equal, the triangle ABC is an isosceles triangle. In an isosceles triangle, the base angles (Angles A and B) are equal, and the sides opposite to those angles (AC and BC) are also equal.
Step 1: Find the measure of Angle C.
We can find the measure of Angle C by subtracting the sum of Angle A and Angle B from 180 degrees.
Angle C = 180 degrees - Angle A - Angle B = 180 degrees - 32 degrees - 32 degrees = 116 degrees.
Step 2: Use trigonometry to find the perpendicular height from C to AB.
We will use the tangent function to find the height, where the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
Using the tangent of Angle C, we have:
tan(C) = height / (AB/2)
Rearranging the formula, we get:
height = (AB/2) * tan(C)
Substituting the values:
height = (108 mm / 2) * tan(116 degrees) = 54 mm * tan(116 degrees).
Step 3: Calculate the value of tan(116 degrees).
tan(116 degrees) ≈ 2.624
Now, substitute the value of tan(116 degrees) back into the previous equation:
height ≈ 54 mm * 2.624 ≈ 141.816 mm.
Therefore, the perpendicular height from C to the base AB is approximately 141.816 mm.
To find the perpendicular height from point C to the base AB in triangle ABC, we can use trigonometry.
Given that angle A = angle B = 32 degrees and AB = 108 mm, we can identify triangle ABC as an isosceles triangle because it has two equal angles. In an isosceles triangle, the two base angles are equal, and the two corresponding sides opposite those angles are also equal.
Since angle A = angle B = 32 degrees, side AC = side BC.
First, let's label the height from point C to the base AB as h.
We can split triangle ABC into two congruent right triangles by drawing a perpendicular line from point C to the base AB.
Now, let's focus on one of the right triangles and apply trigonometric ratios. In this triangle, one of the acute angles is 32 degrees.
Using the sine ratio, we have sin(angle) = opposite/hypotenuse.
In this triangle, the opposite side is the height (h), and the hypotenuse is AC (which is equal to BC).
Therefore, sin(32 degrees) = h/BC.
Let's solve for BC first.
Since triangle ABC is isosceles, we know that AC = BC. Additionally, given AB = 108 mm, we can write:
2 * AC = 108 mm (because AC = BC)
Therefore, AC = 108 mm / 2 = 54 mm.
Now, we can substitute the known values into the equation:
sin(32 degrees) = h/54 mm
To solve for h, we rearrange the equation:
h = sin(32 degrees) * 54 mm
Using a scientific calculator, we can find: sin(32 degrees) ≈ 0.5299.
Therefore:
h ≈ 0.5299 * 54 mm
h ≈ 28.5966 mm (rounded to four decimal places)
So, the perpendicular height from point C to the base AB is approximately 28.5966 mm.