In a triangle, we have <ABC = <ACB = <DEC=<CDE, BC = 8, and DB = 2. Find AB.
How should i approach this? will sss similarity work?
Where do points D and E enter the picture ?
They are inside the triangle ABC
mmmh, and we are supposed to guess where ?
E is on a line that intersects <AB and lies on middle of the line AC, and D is in in the middle of the line BE extending to C.
I need help too
To approach this problem, you can apply the SSS (Side-Side-Side) similarity criterion. However, since the given angles are equal, you can also apply the SAS (Side-Angle-Side) similarity criterion.
Let's use the SAS similarity criterion to solve this problem.
1. Draw a triangle ABC with an angle of 60 degrees at vertex B and an angle of 60 degrees at vertex C. Label the sides as shown:
A
/ \
/ \
B/_____\C
2. Draw a line DE parallel to BC, intersecting AB at point D and intersecting AC at point E.
A
/ \
D /___\E
/ | \
B/___|___\C
3. Given that <ABC = <ACB = <DEC = <CDE = 60 degrees, mark these angles in the diagram.
A
/ \
D /___\E
/ 60|60\
B/____|___\C
4. Since <ABC = <ACB, triangle ABC is an equilateral triangle. Therefore, all sides are equal, and AB = BC = CA = 8.
5. Now, as we have an equilateral triangle ABC, we know that BD and CD bisect the angles at B and C, respectively, in an equilateral triangle.
6. As <ABC = 60 degrees, using the angle bisector theorem, we get that BD/DC = AB/AC. Since BD = 2 and DC = 8, we can substitute these values into the equation.
2/8 = AB/8
Cross-multiplying, we get:
8 × AB = 2 × 8
AB = 2
Therefore, AB = 2.