Question

Основи рівнобічної трапеції дорівнюють 9см і 21 см, а бічна сторона дорівнює 10 см. Знайдіть радіус кола( у см) описаного навколо даної трапеції.

Answer 1

Відповідь:
10,625см

Дано:
Трапеція;
АВ=9см
DC=21см
BC=10см.
R=?
Розв'язання:
ЕС=(DC-AB)/2=(21-9)/2=
=12/2=6см
∆ВЕС- прямокутний трикутник
За теоремою Піфагора:
ВЕ=√(BC²-EC²)=√(10²-6²)=8см.
DE=DC-EC=21-6=15см
∆DBE- прямокутний трикутник.
За теоремою Піфагора
DB=√(DE²+BE)²=√(15²+8²)=
=√(225+64)=√289=17см
р=(10+17+21)/2=48/2=24см, пів-периметра ∆DBC
R=(10*17*21)/(4(√24(24-10)(24-17)(
(24-21))=3570/(4(√(24*14*7*3))=
=3570/(4*84)=3570/336=
=10,625см

Answer 2

To find the radius of the circle circumscribed around the given trapezoid, we can use the formula:

Radius = (Product of the lengths of the diagonals) / (4 times the area of the trapezoid)

Let's calculate the radius step by step:

Step 1: Find the length of the diagonals

The given trapezoid has two diagonals: one connecting the non-parallel sides and the other connecting the midpoints of the parallel sides. Let's calculate their lengths.

The non-parallel sides of the trapezoid are 9 cm and 21 cm, and the parallel sides are 10 cm and 10 cm.

To find the length of the diagonal connecting the non-parallel sides, we can use the Pythagorean theorem. Let's call this diagonal "d1".

Using the Pythagorean theorem: d1^2 = (21 - 9)^2 + 10^2

Simplifying the equation: d1^2 = 12^2 + 10^2

Calculating: d1^2 = 144 + 100 = 244

Taking the square root of both sides: d1 ≈ 15.62 cm

To find the length of the diagonal connecting the midpoints of the parallel sides, we can use the formula for the average of the parallel sides. Let's call this diagonal "d2".

Using the formula: d2 = (10 + 10) / 2 = 10 cm

Step 2: Find the area of the trapezoid

The area of a trapezoid can be calculated using the formula:

Area = (Sum of the lengths of the parallel sides) multiplied by (Height) divided by 2

The parallel sides of the trapezoid are 10 cm and 10 cm, and the height is the length of the non-parallel side, which is 9 cm. Let's calculate the area.

Using the formula: Area = (10 + 10) * 9 / 2 = 90 / 2 = 45 cm^2

Step 3: Calculate the radius

Now that we have the lengths of the diagonals and the area of the trapezoid, we can calculate the radius using the formula:

Radius = (Product of the lengths of the diagonals) / (4 times the area of the trapezoid)

Substituting the values: Radius = (15.62 cm * 10 cm) / (4 * 45 cm^2)

Calculating: Radius ≈ 3.47 cm

Therefore, the radius of the circle circumscribed around the given trapezoid is approximately 3.47 cm.

Please note that the calculations are approximate due to rounding.