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Syllabus

Q.74. In the parallelogram ABCD, M is mid-point of AC and X, Y are points on AB and DC respectively such that AX = CY.

Prove that :

(i) triangle AXM is congruent to triangle CYM,

(ii) KMY is a straight line.

^{0.5}: 1 ( 3^{0.5}= square root of 3)Please do not provide any link.

If AO = 2CO and BO = 2DO; show that:

(i) ∆AOB is similar to ∆COD.

(ii) OA × OD = OB × OC.

(i) the actual length of the diagonal distance AC of the plot in km.

(ii) the actual area of the plot sq km.

Q. In triangle PQR, MN||QR and $\frac{PM}{MQ}=\frac{2}{3}$

(i) Find $\frac{MN}{QR}$

(ii) Prove that triangle OMN and ORQ are similar

(iiI) Find, area of triangle OMN : area of triangle ORQ

1. the actual length of the diagonal distance AC of the plot in km.

2. the actual area of the plot in sq km.

(i) the length of PQ, if BC=7.5cm.

(ii)the area of triangle APQ: area of triangle ABC

(iii)the area of triangleAPQ:area of PBCQ

(b) In $\u2206$PQR, MN is parallel to QR and

$\frac{PM}{MQ}=\frac{2}{3}$

(i) Find

$\frac{MN}{QR}$

(ii) Prove that $\u2206$OMN and

$\u2206$ORQ are similar.

(iii) Find , Area of $\u2206$OMN : Area of $\u2206$ORQ

$14.PistheorthocenterOf\u2206ABC.ShowthatAistheorthocenterof\u2206PBC.\phantom{\rule{0ex}{0ex}}15.Histheorthocentreof,\u2206ABCX.YandZarerespetiveIythemid-pointsofAH,BH\phantom{\rule{0ex}{0ex}}andCHshowthatHisalsotheof\u2206XYZ.$

EXAMPLE 10 In Fig. 8.29, if x + y = w + z, then prove that AOB is a line.

SOLUTION Since sum of all the angles round a point is equal to 360$\xb0$. Therefore,

2. Prove that the figure obtained by joining the mid-points of the adjacent side of a rectangle is a rhombus.

3.D, E and F are the mid-points of the side AB, BC and CA of an isosceles $\u25b3$ ABC in which AB=BC. Prove that $\u25b3$DEF is also issosceles.