- In figure 5.38, points X, Y, Z are the midpoints of side AB, side BC and side AC of…
- In figure 5.39, PQRS and MNRL are rectangles. If point M is the midpoint of side PR…
- In figure 5.40, ΔABC is an equilateral triangle. Points F,D and E are midpoints of side…
- In figure 5.41, seg PD is a median of ΔPQR, Point T is the midpoint of seg PD. Produced…
Practice Set 5.5
Question 1.In figure 5.38, points X, Y, Z are the midpoints of side AB, side BC and side AC of ΔABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
Answer:
Given X , Y and Z is the mid-point of AB, BC and AC.
Length of AB = 5 cm
So length of ZY = 1/2 × AB =1/2 × 5=2.5 cm (line joining mid-point of two sides of a triangle is parallel of the third side and is half of it)
Similarly, XZ = 1/2 × BC = 1/2 × 11= 5.5cm
Similarly, XY = 1/2 × AC = 1/2 × 9 = 4.5cm
Question 2.
In figure 5.39, PQRS and MNRL are rectangles. If point M is the midpoint of side PR then prove that,
i. SL = LR. Ii. LN = 1/2SQ.
Answer:
The two rectangle PQRS and MNRL
In Δ PSR,
∠ PSR = ∠ MLR = 90°
∴ ML ∥ SP when SL is the transversal
M is the midpoint of PR (given)
By mid-point theorem a parallel line drawn from a mid-point of a side of a Δ meets at the Mid-point of the opposite side.
Hence L is the mid-point of SR
⇒ SL= LR
Similarly if we construct a line from L which is parallel to SR
This gives N is the midpoint of QR
Hence LN∥ SQ and L and N are mis points of SR and QR respectively
And LN = 1/2 SQ (mid-point theorem)
Question 3.
In figure 5.40, ΔABC is an equilateral triangle. Points F,D and E are midpoints of side AB, side BC, side AC respectively. Show that ΔEFD is an equilateral triangle.
Answer:
Given F, D and E are mid-point of AB, BC and AC of the equilateral ΔABC ∴ AB =BC = AC
So by mid-point theorem
Line joining mid-points of two sides of a triangle is 1/2 of the parallel third side.
∴ FE = 1/2 BC =
Similarly, DE = 1/2 AB
And FD = 1/2 AC
But AB =BC = AC
⇒ 1/2 AB = 1/2 BC = 1/2 AC
⇒ DE = FD = FE
Since all the sides are equal ΔDEF is a equilateral triangle.
Question 4.
In figure 5.41, seg PD is a median of ΔPQR, Point T is the midpoint of seg PD. Produced QT intersects PR at M. Show that
[Hint : draw DN || QM.]
Answer:
PD is median so QD = DR (median divides the side opposite to vertex into equal halves)
T is mid-point of PD
⇒ PT = TD
In ΔPDN
T is mid-point and is ∥ to TM (by construction)
⇒TM is mid-point of PN
PM =MN……………….1
Similarly in ΔQMR
QM ∥ DN (construction)
D is mid –point of QR
⇒ MN = NR…………………..2
From 1 and 2
PM = MN = NR
Or PM = 1/3 PR
⇒ hence proved