The diagonal of a quadrilateral measures 24 cm and the lengths of the perpendiculars drawn to it from the opposite vertices is in the ratio 3 : 2. Find the lengths of the perpendiculars, if the area of the quadrilateral is 390 m².
Answers
Step-by-step explanation:
Area of the field
= Area of upper triangle + Area of Lower triangle
=
2
1
×13×24+
2
1
×8×24
=156+96=252 sq m
Given :
- Area of Quadrilateral = 390m²
- Diagonal of Quadrilateral = 24cm
- Ratio of the two perpendiculars = 3:2
To Find :
- The lengths of the perpendiculars.
Solution :
✰ As we know that, Area of Quadrilateral is given by ½ × diagonal × (sum of perpendiculars). Now in this question, Area of Quadrilateral and Diagonal of Quadrilateral along with Ratio of perpendiculars are given. So we will assume the first and second perpendicular as 3x and 2x respectively . After that we will simply put the given values in the formula to find the Perpendiculars of the Quadrilateral.
⠀⠀
⠀⟼⠀⠀Area = ½ × diagonal × (P1 + P2)
⠀⟼⠀⠀390 = ½ × 24 × (3x + 2x)
⠀⟼⠀⠀390 = ½ × 24 × 5x
⠀⟼⠀⠀390 × 2 = 24 × 5x
⠀⟼⠀⠀780 = 24 × 5x
⠀⟼⠀⠀780/24 = 5x
⠀⟼⠀⠀32.5 = 5x
⠀⟼⠀⠀32.5/5 = x
⠀⟼⠀⠀6.5m = x
⠀⠀
Therefore :
⇢ First Perpendicular = 3x
⇢ First Perpendicular = 3 × 6.5
⇢ First Perpendicular = 19.5m
⠀
⇢ Second Perpendicular = 2x
⇢ Second Perpendicular = 2 × 6.5
⇢ Second Perpendicular = 13m
⠀⠀
Thus Perpendicular of the Quadrilateral are 19.5m and 13m respectively.
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