Mathematics
In the given figure, LMN is a right triangle in which ∠M = 90°, P and Q are mid-points of LM and LN respectively. If LM = 9 cm, MN = 12 cm and LN = 15 cm, find :
(i) the perimeter of trapezium MNQP
(ii) the area of trapezium MNQP
Mid-point Theorem
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Answer
Since, P is mid-point of LM,
LP = PM
From figure,
⇒ LM = LP + PM
⇒ 9 = PM + PM
⇒ 9 = 2 PM
⇒ PM =
⇒ PM = 4.5 cm
Since, Q is mid-point of LN,
LQ = QN
From figure,
⇒ LN = LQ + QN
⇒ 15 = QN + QN
⇒ 15 = 2 QN
⇒ QN =
⇒ QN = 7.5 cm
By mid-point theorem,
The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
Since, P and Q are the mid-points of LM and LN respectively.
⇒ PQ = × 12 = 6 cm
(i) Perimeter of trapezium MNQP = PM + PQ + QN + MN
⇒ 4.5 + 6 + 7.5 + 12
⇒ 30 cm.
Hence, perimeter of trapezium MNQP = 30 cm.
(ii) Area of trapezium MNQP = × (Sum of parallel sides) × height of trapezium
= × (PQ + MN) × PM
= × (6 + 12) × 4.5
= × 18 × 4.5
= 40.5 cm2
Hence, area of trapezium MNQP is 40.5 cm2.
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