In an isosceles trapezoid, the perimeter is 100, and the area is 500, you can inscribe a circle.

| September 23, 2021 | education

In an isosceles trapezoid, the perimeter is 100, and the area is 500, you can inscribe a circle. Find the distance from the point of intersection of the diagonals of the trapezoid to its smaller base.

Since a circle can be inscribed in a trapezoid, then (AB + CD) = (BC + AD) = Ravsd / 2 = 100/2 = 50 cm.

Since the trapezoid is isosceles, then AB = CD = 50/2 = 25 cm.

Let’s build the height of the HВ.

The area of ​​the trapezoid is equal to: Savsd = (ВС + АD) * ВН / 2.

500 = 50 * СН / 2.

BH = 1000/50 = 20 cm.

In a right-angled triangle ABН, according to the Pythagorean theorem, AH ^ 2 = AB ^ 2 – BH ^ 2 = 625 – 400 = 225.

AH = 15 cm.

Let’s build the height of the CP, then DP = AH = 15 cm, and BC = HP.

Then AH + HP + DP + BC = 50.

2 * BC = 50 – 15 – 15 = 20.

BC = 20/2 = 10 cm.

AD = 50 – 10 = 40 cm.

Let’s build the CM height.

Triangles KOС and MOD are rectangular and similar in acute angle, KC = BC / 2 = 5 cm, DM = AD / 2 = 20 cm.

Then OK / ОМ = KС / DМ = 5/20 = 1/4.

ОМ = 4 * OK.

OM + OK = KM = 20 cm.

5 * OK = 20.

OK = 20/5 = 4 cm.

Answer: The distance from the point of intersection of the diagonals to the smaller base is 4 cm.



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