In triangle DEF it is known that angle D = 90, angle F = 30. The bisector of angle E intersects

In triangle DEF it is known that angle D = 90, angle F = 30. The bisector of angle E intersects the leg DF at point P. Find FP if EP + PD = 12cm.

Given:
triangle DEF,
angle D = 90, angle F = 30,
EP – bisector,
EP + PD = 12 cm.
Find FP length -?
Solution:
1) Consider the triangle DEF.
angle E = 180 – (angle D + angle F);
angle E = 180 – (90 + angle 30);
angle E = 180 – 120;
angle E = 60;
2) Since EP is a bisector, the angle DEP = PEF = 60: 2 = 30;
3) Consider a right-angled triangle DEP. Opposite the angle of 30 degrees lies the leg, which is equal to half of the hypotenuse, that is, DP = 1/2 * EP;
2) Since EP + PD = 12 cm, then
EP + 1/2 EP = 12;
Ep * (1 + 1/2) = 12;
EP * 1 1/2 = 12;
EP = 12: 1 1/2;
EP = 12: 3/2;
EP = 12 * 2/3;
EP = (12 * 2) / 3;
EP = (4 * 2) / 1;
EP = 8 cm.
Answer: 8 centimeters.