The area of a quadrilateral is 227.2 cm² and the length of the perpendiculars…
2019
The area of a quadrilateral is 227.2 cm² and the length of the perpendiculars from the opposite vertices to a diagonal are 7.2 cm and 8.8 cm. What is the length of the diagonal?
Answer: B. 28.4 cm — When a diagonal is drawn in any quadrilateral, it splits the quadrilateral into two triangles that share that diagonal as a common base. The quadrilateral's…
- A.
26.8 cm
- B.
28.4 cm
- C.
30.2 cm
- D.
32.6 cm
Show answer & explanation
Correct answer: B
When a diagonal is drawn in any quadrilateral, it splits the quadrilateral into two triangles that share that diagonal as a common base. The quadrilateral's area therefore equals half the diagonal's length multiplied by the sum of the perpendicular distances dropped onto that diagonal from the two opposite vertices: Area = 1/2 × d × (h1 + h2), where d is the diagonal and h1, h2 are the two perpendiculars.
Applying this relation to the given quadrilateral:
Add the two perpendicular distances: 7.2 cm + 8.8 cm = 16 cm.
Substitute the given area and this sum into the formula: 227.2 = 1/2 × d × 16.
Simplify the right-hand side: 227.2 = 8 × d.
Divide both sides by 8 to isolate d: d = 227.2 / 8 = 28.4 cm.
As an independent check, substituting d = 28.4 cm back into the formula gives 1/2 × 28.4 × 16 = 227.2 cm², which exactly matches the quadrilateral's given area -- confirming the diagonal measures 28.4 cm.