Everything Maths Grade 10

5.Dis a point onBC, in△ABC. Nis the mid-point ofAD.Ois the mid-point ofABandMis the
mid-point ofBD.N R∥AC.

B

O

C

N

A

D

M

R

a)Prove thatOBM Nis a parallelogram.

b)Prove thatBC= 2M R.

6.In△M N P,M^= 90°,Sis the mid-point ofM NandTis the mid-point ofN R.

M

T

N U P

S

R

a)ProveUis the mid-point ofN P.

b)IfST=4 cm and the area of△SN Tis 6 cm^2 , calculate the area of△M N R.

c)Prove that the area of△M N Rwill always be four times the area of△SN T, letST=xunits and

SN=yunits.

7. a)Given quadrilateralQRSTwith sidesQR∥T SandQT∥RS. Also given:Q^=yandS^= 63◦;
   QT R^ = 38◦andRT S^ =x. Complete the proof below to prove thatQRSTis a parallelogram.

Q

R S

T

y

63 ◦

x

38 ◦

Steps Reasons

QT R^ =TRS^ alt\s;QT∥RS

ST R^ =QRT^ alt\s;QR∥T S

??

)△QRT△ST R (AAS)

? congruent triangles

Q^=S^ congruent triangles

)QRSTis a parallelogram?

b)Calculate the value ofy.

c)Calculate the value ofx.

Chapter 12. Euclidean geometry 411