The solutions using the relevant theorems are:
13. x = −3 or x = 4
14. x = 12
15. x = 12
16. x = 1/2 or x = 4
17. x = 5
18. x = −1 or x = 6
What is the Triangle Midsegment Theorem?
The midsegment of a triangle that joins two sides of a triangle divides the two sides proportionally based on the triangle midsegment theorem.
What is the Angle Bisector Theorem?
The angle bisector of a triangle, according to the angle bisector theorem divides the opposite angles sides in a proportional manner.
What is the Parallel Lines and Proportionality Theorem?
The theorem states that when three or more lines that are parallel is cut across by two transversals, they are divided by the transversal proportionally.
13. Apply the angle bisector theorem:
x/3 = (x + 4)/(x + 2)
Cross multiply
x(x + 2) = 3(x + 4)
x² + 2x = 3x + 12
x² + 2x - 3x - 12 = 0
x² - x - 12 = 0
Factorize
x = −3 or x = 4
14. Apply the angle bisector theorem:
16/x = 12/9
x(12) = (16)(9)
12x = 144
x = 12
15. Apply the angle bisector theorem:
(x - 4)/6 = x/9
9(x - 4) = 6x
9x - 36 = 6x
9x - 6x = 36
3x = 36
x = 12
16. Apply the triangle midsegment theorem:
3x/(x + 4) = (x - 1)/(x - 2)
Cross multiply
3x/(x + 4) = (x - 1)/(x - 2)
(x - 1)(x + 4) = 3x(x - 2)
x² + 3x - 4 = 3x² - 6x
x² + 3x - 4 - 3x² + 6x = 0
-2x² + 9x - 4 = 0
Factorize
x = 1/2 or x = 4
17. Apply the Parallel Lines and Proportionality Theorem:
(x + 3)/(2x + 2) = (2x + 2)/(4x - 2)
Cross multiply
4x² + 10x - 6 = 4x² + 8x + 4
4x² + 10x - 6 - 4x² - 8x - 4 = 0
2x - 10 = 0
2x = 10
x = 5
18. Apply the angle bisector theorem:
(2x + 3)/x = (x + 4)/(x - 2)
Cross multiply
(2x + 3)(x - 2) = x(x + 4)
2x² - x - 6 = x² + 4x
2x² - x - 6 - x² - 4x = 0
x² - 5x - 6 = 0
Factorize
x = −1 or x = 6
Learn more about the angle bisector theorem on:
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