MO Z9–I–2 - 2017

In the VODY trapezoid, VO is a longer base and the diagonal intersection K divides the VD line in a 3:2 ratio. The area of the KOV triangle is 13.5 cm². Find the area of the entire trapezoid.

Correct result:

S =  37.5 cm²

Solution:

$$\begin{gathered} S_1=13.5{\text{cm}}^2 \\ {k}_2=(2\mathrm{/}3{)}^2={\displaystyle \frac{4}{9}}\doteq 0.4444 \\ {S}_2=k_2\cdot S_1=0.4444\cdot 13.5=6{\text{cm}}^2 \\ {k}_3=(2+3)\mathrm{/}3={\displaystyle \frac{5}{3}}\doteq 1.6667 \\ {S}_3=k_3\cdot S_1-S_1=1.6667\cdot 13.5-13.5=9{\text{cm}}^2 \\ {k}_4=(2+3)\mathrm{/}2={\displaystyle \frac{5}{2}}=2.5 \\ {S}_4=k_4\cdot S_2-S_2=2.5\cdot 6-6=9{\text{cm}}^2 \\ S=S_1+S_2+S_3+S_4=13.5+6+9+9={\displaystyle \frac{75}{2}}=37.5{\text{cm}}^2 \end{gathered}$$

Try another example

Showing 0 comments:

Next similar math problems:

• Trapezoid - hard example
  Base of the trapezoid are: 24, 16 cm. Diagonal 22, 26 cm. Calculate its area and perimeter.
• See harmonics
  It is true that the size of the central segment of any trapezoid is the harmonic mean size of its bases? Prove it. Central segment crosses the intersection of the diagonals and is parallel to the bases.
• Diagonal intersect
  isosceles trapezoid ABCD with length bases | AB | = 6 cm, CD | = 4 cm is divided into 4 triangles by the diagonals intersecting at point S. How much of the area of the trapezoid are ABS and CDS triangles?
• Isosceles trapezoid
  In an isosceles trapezoid KLMN intersection of the diagonals is marked by the letter S. Calculate the area of trapezoid if /KS/: /SM/ = 2:1 and a triangle KSN is 14 cm².
• Trapezoid - diagonal
  Trapezoid has a length of diagonal AC corssed with diagonal BD in the ratio 2:1. The triangle created by points A, cross point of diagonals S and point D has area 164 cm². What is the area of the trapezoid?
• Trapezoid IV
  In a trapezoid ABCD (AB||CD) is |AB| = 15cm |CD| = 7 cm, |AC| = 12 cm, AC is perpendicular to BC. What area has a trapezoid ABCD?
• Trapezoid - central median
  The central median divides the trapezoid into two smaller trapezoids. Determines the ratio of their contents.
• Isosceles trapezoid
  Isosceles trapezoid ABCD, AB||CD is given by |CD| = c = 12 cm, height v = 16 cm and |CAB| = 20°. Calculate area of the trapezoid.
• Tree shadow
  The shadow of the tree is 16 meters long. Shadow of two meters high tourist sign beside standing is 3.2 meters long. What height has tree (in meters)?
• Divide
  Divide area of rectangles with dimensions 32m and 10m by the ratio 7: 9. What area corresponds to a smaller section?
• Median
  The median of the triangle LMN is away from vertex N 84 cm. Calculate the length of the median, which start at N.
• Numbers at ratio
  The two numbers are in a ratio 3:2. If we each increase by 5 would be at a ratio of 4:3. What is the sum of original numbers?
• The farmer
  The farmer would like to first seed his small field. The required amount depends on the seed area. Field has a triangular shape. The farmer had fenced field, so he knows the lengths of the sides: 119, 111 and 90 meters. Find a suitable way to determine th
• The angles
  The angles in the triangle are in the ratio 12: 15: 9. Find the angles.
• Similarity coefficient
  The triangles ABC and A "B" C "are similar to the similarity coefficient 2. The sizes of the angles of the triangle ABC are α = 35° and β = 48°. Find the magnitudes of all angles of triangle A "B" C ".
• Centre of mass
  The vertices of triangle ABC are from the line p distances 3 cm, 4 cm and 8 cm. Calculate distance from the center of gravity of the triangle to line p.
• Rectangular triangles
  The lengths of corresponding sides of two rectangular triangles are in the ratio 2:5. At what ratio are medians relevant to hypotenuse these right triangles? At what ratio are the contents of these triangles? Smaller rectangular triangle has legs 6 and 8