Less than the base of an isosceles trapezoid is 8 cm, the lateral side is 10 cm and an acute angle

Less than the base of an isosceles trapezoid is 8 cm, the lateral side is 10 cm and an acute angle at the base of 45, find the perimeter of this isosceles trapezoid.

Given:
ABCE – isosceles trapezoid,
BC = 8 centimeters,
AB = CE = 10 centimeters,
angle A = angle E = 45 degrees.
Find the perimeter of the isosceles trapezoid ABCE, that is, P ABCE -?
Decision:
1. Drop perpendiculars BO and CH.
2. Rectangular triangles ABO = COE by hypotenuse and acute angle, since AB = CE, angle A = angle E.
3. The BCHO quadrilateral is rectangles. Then BC = OH = 8 centimeters.
4. Consider a right-angled triangle ABO. AVO angle = 180 – 90 – 45 = 45 (degrees). Then the right-angled triangle ABO is also isosceles, then AO = OB.
By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AO ^ 2 + BO ^ 2 = AB ^ 2;
AO ^ 2 + AO ^ 2 = 10 ^ 2;
2 * AO ^ 2 = 100;
AO ^ 2 = 100: 2;
AO ^ 2 = 50;
AO = 5√2 centimeters;
5. AE = AO + OH + NOT;
AE = 5√2 + 8 + 5√2;
AE = 10√2 + 8;
AE = 18√2.
6. R ABCE = 10 + 8 + 10 + 18√2;
P ABCE = 46√2 centimeters.
Answer: 46√2 centimeters.