The Mid-point Theorem | Quadrilaterals | Assessment | English | Grade 9

As per the midpoint theorem, In ΔABC, if point P is the midpoint of side AB and point Q is the midpoint of side AC, then

• PQ || BC & PQ = BC

• PQ || BC & PQ = ½ BC

• PQ || BC

• PQ || BC & ½ PQ = ½ BC

Answer explanation

As per the midpoint theorem, the line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side.

Tags

CCSS.HSG.SRT.B.4

Points E and F are mid points of side AB and side AC of ΔABC respectively. If BC = 24 cm then find the length of EF.

• 48

• 36

• 42

• 12

Answer explanation

In ABC, point E and F are midpoints of side AB and side AC respectively. EF = ½ BC ----- by midpoint theorem EF = ½ x 24 EF = 12 cm

Tags

CCSS.HSG.SRT.B.4

In the given figure, ABCD and APQR are parallelograms. With the help of the given figure, find the perimeter of parallelogram ABCD.

• 48

• 72

• 24

• 36

Answer explanation

In ΔAPQ, AB = BP and AC = CQ --- given ∴ BC = ½ PQ --- by midpoint theorem BC = ½ x 10 --- PQ = 10 cm ∴ BC = 5 cm Similarly, in Δ ARQ, AD = DR and AC = CQ --- given ∴ DC = ½ RQ --- by midpoint theorem DC = ½ x 14--- RQ = 14 cm ∴ DC = 7 cm In 􀀍ABCD, Perimeter of a parallelogram = 2(l + b) Perimeter of 􀀍ABCD = 2 (5 + 7) Perimeter of 􀀍ABCD = 24 cm

Tags

CCSS.HSG.CO.C.11

ΔABC is an equilateral triangle. Points F, D and E are midpoints of side AB, side BC, side AC respectively. What is the type of ΔFED?

• Right angled triangle

• Equilateral Triangle

• Isosceles Triangle

• Scalene Triangle

Answer explanation

In ΔABC, It is given that Points F, D and E are midpoints of side AB, side BC and side AC respectively. By midpoint theorem, FD = ½ BC ---- 1 FE = ½ AC ---- 2 DE = ½ AB ---- 3 But, AB = AC = BC ---- Because it is given that ΔABC is an equilateral triangle, ∴ FD = FE = DE --- From 1, 2 and 3 Hence, ΔFED is an equilateral triangle.

Tags

CCSS.HSG.CO.C.10

In figure, 􀀍PQRS and 􀀍YZRX are rectangles. If point Y is the midpoint of side PR then prove (i) SX = XR (ii) XZ = ½ SQ Click 'Yes' after completing this question.

• Yes

• No

Answer explanation

Proof: In 􀀍YZRX, XY || RZ --- Opposite sides of a rectangle are parallel …. (1) In 􀀍PQRS, SP || RQ --- Opposite sides of a rectangle are parallel We can also say that, SP || RZ --- R – Z – Q …. (2) From (1) & (2), SP || XY … (3) In ΔSRP, RY = PY --- Y is the midpoint of PR and SP || XY … from (3) Hence, X will be the midpoint of SR --- The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side. i.e. SX = XR …(4) Similarly we can prove that, RZ = ZQ … (5) In ΔSRQ, Diagram Point X is the midpoint of SR … from 4 Point Z is the midpoint of RQ … from 5 Hence, XZ = ½ SQ

Tags

CCSS.HSG.CO.C.11

A figure obtained by joining the midpoints of a rhombus will be a _______

• Square

• Rectangle

• Trapezium

• Rhombus

Answer explanation

Diagram Given: □ABCD is a Rhombus To Prove: □PQRS is a rectangle Construction: Join AC, PR and SQ In ΔABC, P is the midpoint of AB and Q is the midpoint of BC. PQ ∣∣ AC PQ = ½ AC ---- by using mid-point theorem ---1 Similarly, in ΔDAC, SR ∣∣ AC SR = ½ AC ---- by using mid-point theorem ---2 From equation 1 and 2, PQ ∣∣ SR PQ = SR So, □PQRS is a parallelogram. Similarly □ABQS is a parallelogram. AB = SQ ----- opposite sides of a parallelogram are equal --- 3 Similarly, □PBCR is a parallelogram. BC = PR ---- opposite sides of a parallelogram are equal ---- 4 AB = PR --- ∵BC = AB sides of a rhombus SQ = PR ------ from Eq. 3 So, the diagonals of a parallelogram are equal. Hence,□PQRS is a rectangle.

Tags

CCSS.HSG.GPE.B.6