In triangle ABC, BC = 20sqrt3 and angle C = 30 degrees. Let the perpendicular bisector of BC intersect BC and AC at D and E respectively. Find the length of E.
Since triangle DEC is a right triangle, with angle C = 30° and DC=10√3, DE = 10.
No idea what the "length" of E is, since it's a point.
To find the length of E, we can use the fact that the perpendicular bisector of a side of a triangle divides that side into two equal lengths.
Given that BC = 20√3, the length of BD is half of BC. Therefore, BD = 20√3 / 2 = 10√3.
Since the perpendicular bisector of BC intersects BC and AC at D and E, respectively, DE is the perpendicular bisector of BC.
Since DE is the perpendicular bisector of BC, it divides BC into two equal lengths. Therefore, DE = BD = 10√3.
Hence, the length of E is 10√3.
To find the length of E, we need to use some geometry concepts related to perpendicular bisectors and triangle properties. Here are the steps to solve this problem:
1. Draw triangle ABC with side BC = 20sqrt3 and angle C = 30 degrees.
2. Draw the perpendicular bisector of BC. The perpendicular bisector is a line that divides BC into two equal halves and is perpendicular to BC. Let's call the point where the perpendicular bisector intersects BC as point D.
3. Since the perpendicular bisector is also a line of symmetry, AD will also be the perpendicular bisector of AC. Therefore, DB = DC = AB = AC.
4. Since angle C is 30 degrees, angle DCA is also 30 degrees. Hence, triangle DCA is an equilateral triangle.
5. In an equilateral triangle, all sides are equal. So, AD = AC = 20sqrt3.
6. Now, we need to find the length of DE. To do that, we need to find the length of AE.
7. Since AD is the perpendicular bisector of AC, DE is also half of AC. Therefore, DE = AE = (1/2) * 20sqrt3 = 10sqrt3.
Hence, the length of E is equal to 10sqrt3.