Rhombus and inscribed circle

It is given a rhombus with side a = 6 cm and the radius of the inscribed circle r = 2 cm. Calculate the length of its two diagonals.

Correct answer:

u =  11.2101 cm
v =  4.2819 cm

Step-by-step explanation:

$$\begin{gathered} a=6\text{ cm } \\ r=2\text{ cm } \\ a=a_1+a_2 \\ {r}^2=a_1{a}_2 \\ {a}_1(a-a_1)=r^2 \\ x(6-x)=2^2 \\ x(6-x)=2^2 \\ -x^2+6x-4=0 \\ {x}^2-6x+4=0 \\ a=1;b=-6;c=4 \\ D=b^2-4ac=6^2-4\cdot 1\cdot 4=20 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{6\pm \sqrt{20}}{2}}={\displaystyle \frac{6\pm 2\sqrt{5}}{2}} \\ {x}_{1,2}=3\pm 2.2360679775 \\ {x}_1=5.2360679775 \\ {x}_2=0.7639320225 \\ \text{ Factored form of the equation: } \\ (x-5.2360679775)(x-0.7639320225)=0 \\ {a}_1=x_1=5.2361\doteq 5.2361 \\ {a}_2=x_2=0.7639\doteq 0.7639 \\ (u\mathrm{/}2{)}^2=a_1^2+r^2 \\ u=2\cdot \sqrt{a_1^2+r^2}=2\cdot \sqrt{5.2361^2+2^2}=11.2101\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} (v\mathrm{/}2{)}^2=a_2^2+r^2 \\ v=2\cdot \sqrt{a_2^2+r^2}=2\cdot \sqrt{0.7639^2+2^2}\doteq 4.2819\text{ cm } \\ {r}_2=\sqrt{a_1\cdot a_2}=\sqrt{5.2361\cdot 0.7639}=2 \\ {r}_2=r \end{gathered}$$

Try another example

Showing 1 comment:

Math student

The diagonal of a rhombus measure 16cm and 30cm find its perimeter

4 years ago  Like

Related math problems and questions:

• Rhombus
  It is given a rhombus of side length a = 19 cm. Touchpoints of inscribed circle divided his sides into sections a₁ = 5 cm and a₂ = 14 cm. Calculate the radius r of the circle and the length of the diagonals of the rhombus.
• Diagonals
  A diagonal of a rhombus is 20 cm long. If it's one side is 26 cm, find the length of the other diagonal.
• Rhombus IV
  Calculate the length of the diagonals of the rhombus, whose lengths are in the ratio 1: 2 and a rhombus side is 35 cm.
• Tangents
  To circle with a radius of 41 cm from the point R guided two tangents. The distance of both points of contact is 16 cm. Calculate the distance from point R and circle centre.
• Diagonals
  Given a rhombus ABCD with a diagonalsl length of 8 cm and 12 cm. Calculate the side length and content of the rhombus.
• Rhombus
  Calculate the perimeter and area of ​​a rhombus whose diagonals are 39 cm and 51 cm long.
• A square
  A square with a length of diagonals 12cm give: a) Calculate the area of a square b) rhombus with the same area as the square, has one diagonal with a length of 16 cm. Calculate the length of the other diagonal.
• Rhombus and diagonals
  The lengths of the diamond diagonals are e = 48cm, f = 20cm. Calculate the length of its sides.
• Diamond diagonals
  Calculate the diamonds' diagonals lengths if the diamond area is 156 cm square, and the side length is 13 cm.
• Rhombus OWES
  OWES is a rhombus given that OW 6cm and one diagonal measures 8cm. Find its area?
• Area of iso-trap
  Find the area of an isosceles trapezoid if the lengths of its bases are 16 cm and 30 cm, and the diagonals are perpendicular to each other.
• Diagonals in diamons/rhombus
  Rhombus ABCD has side length AB = 4 cm and a length of one diagonal of 6.4 cm. Calculate the length of the other diagonal.
• Diamond
  Side length of diamond is 35 cm and the length of the diagonal is 56 cm. Calculate the height and length of the second diagonal.
• Logs
  The trunk diameter is 52 cm. Is it possible to inscribe a square prism with side 36 cm?
• Rectangle diagonals
  It is given a rectangle with an area of 24 cm² a circumference of 20 cm. The length of one side is 2 cm larger than the length of the second side. Calculate the length of the diagonal. Length and width are yet expressed in natural numbers.
• Ratio of sides
  Calculate the area of a circle with the same circumference as the circumference of the rectangle inscribed with a circle with a radius of r 9 cm so that its sides are in ratio 2 to 7.
• RT - inscribed circle
  In a rectangular triangle has sides lengths> a = 30cm, b = 12.5cm. The right angle is at the vertex C. Calculate the radius of the inscribed circle.