Mathematics
In which of the following cases you will get 2XY = QR for the given figure?
(i) When PX = QX and PY = RY
(ii) When PX = QX and a + b = 180°
Only in case (i)
Only in case (ii)
In both the cases
None of these
Mid-point Theorem
2 Likes
Answer
In case (i) :
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 is equal to half of it.
In △PQR,
Given,
PX = QX and PY = RY
⇒ X and Y are midpoints of PQ and PR respectively.
∴ XY || QR and XY = QR
⇒ QR = 2 XY
∴ Case (i) is true.
In case (ii) :
Given,
PX = QX
⇒ X is the mid-point of PQ
By converse of mid-point theorem,
A line drawn through the midpoint of one side of a triangle, and parallel to another side, will bisect the third side.
Given,
⇒ ∠QXY + ∠XQR = 180°
⇒ a + b = 180°
⇒ ∠QXY and ∠XQR are co-interior angles and their sum is equal to 180°.
∴ XY is parallel to QR.
In △PQR,
Since, X is the mid-point of PQ and XY // QR, thus :
Y is mid-point of PR.
Since, X and Y are mid-points of side PQ and PR respectively.
⇒ XY = QR (By mid-point theorem)
⇒ QR = 2 XY
∴ Case (ii) is true.
Hence, option 3 is the correct option.
Answered By
3 Likes
Related Questions
The figure formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order is a square only, if :
ABCD is a rhombus
Diagonals of ABCD are equal
Diagonals of ABCD are equal and perpendicular
Diagonals of ABCD are perpendicular
D and E are the mid-points of the sides AB and AC respectively of △ABC. DE is produced to F. To show that CF is equal and parallel to DA, we need an additional information, which is :
DE = EF
AE = EF
∠DAE = ∠EFC
∠ADE = ∠ECF
In the figure, R is the mid-point of AB, P is the mid-point of AR and L is the mid-point of AP. If RS, PQ and LM are parallel to each other, then the length of BC is :
3 LM
4 LM
6 LM
8 LM
In the given figure, △ABC is a scalene triangle in which ∠B = 90°. P is the mid-point of AB, PQ || BC and QM ⊥ BC. Which type of quadrilateral is PQMB?
