2. 12. Sides AB and BC and median AD of triangle ABC are respectively proportional to sides PQ and
QR and median PM of ΔPQR (see figure). Show that ΔABC ~ ΔPQR.
Sol. We have ΔABC and ΔPQR in which AD and PM are medians are propotional to corresponding sides
BC and QR respectively such, that
𝐴𝐵
𝑃𝑄
=
𝐵𝐶
𝑄𝑅
=
𝐴𝐷
𝑃𝑀
𝐴𝐵
𝑃𝑄
=
1
2
𝐵𝐶
1
2 𝑄𝑅
=
𝐴𝐷
𝑃𝑀
𝐴𝐵
𝑃𝑄
=
𝐵𝐷
𝑄𝑀
=
𝐴𝐷
𝑃𝑀
∴Using SSS similarity, we have:
Their corresponding q es are equal
⇒∠ABD = ∠PQM
∴∠ABC = ∠PQR
Now, in ∆ABC and ∆PQR
𝐴𝐵
𝑃𝑄
=
𝐵𝐶
𝑄𝑅
……………(i) Given
∠ABC = ∠PQR ………..(ii) proved above
ΔABC ~ ΔPQR by SAS similarity criteria
3. 13. D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC. Show that CA2 = CB . CD.
Sol. We have a ΔABC and point D on its side BC such
that ∠ADC = ∠BAC
In ΔABC and ΔADC
∠BAC = ∠ADC [Given]
And ∠BCA = ∠DCA
∴ Using AA similarity, we have
Δ BAC ~ Δ ADC
∴Their corresponding sides are proportional
4. 14. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR
and median PM of another ,triangle PQR. Show that ΔABC ~ ΔPQR.
Sol. We have two Δ’s ABC and PQR such that AD and PM are medians proportional to corresponding sides
BC and QR respectively. Also
𝐴𝐵
𝑃𝑄
=
𝐵𝐶
𝑄𝑅
=
𝐴𝐷
𝑃𝑀
𝐴𝐵
𝑃𝑄
=
1
2
𝐵𝐶
1
2 𝑄𝑅
=
𝐴𝐷
𝑃𝑀
𝐴𝐵
𝑃𝑄
=
𝐵𝐷
𝑄𝑀
=
𝐴𝐷
𝑃𝑀
since, the corresponding angles of similar triangles are equal.
∴∠ABD = ∠PQM
⇒ ∠ABC = ∠PQR ...(2)
Now, in ΔABC and ΔPQR
∠ABC = ∠PQR [From (2)]
𝐴𝐵
𝑃𝑄
=
𝐵𝐶
𝑄𝑅
By SAS ΔABC ~ ΔPQR.
5. 15. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a
tower casts a shadow 28 m long. Find the height of the tower.
Sol. Let AB = 6 m be the pole and BC = 4 m be its shadow (in right
ΔABC), whereas DE and EF denote the tower and its shadow
respectively.
EF = Length of the shadow of the tower = 28 m
And DE = h = Height of the tower
In ΔABC and ΔDEF we have
∠B = ∠E = 90°
∠A = ∠D [Angular elevation of the sun at the same time].
∴Using AA criteria of similarity, we have
ΔABC ~ ΔDEF
∴Their sides are proportional
Thus, the required height of the tower is 42 m.
6. 16. If AD and PM are medians of triangles ABC and PQR, respectively where ΔABC ~ ΔPQR, prove that
𝐴𝐵
𝑃𝑄
=
𝐴𝐷
𝑃𝑀
Sol. We have ΔABC ~ ΔPQR such that AD and PM are the medians.
∵ΔABC ~ ΔPQR
And the corresponding sides of similar triangles are proportional.
𝐴𝐵
𝑃𝑄
=
𝐵𝐶
𝑄𝑅
=
𝐴𝐶
𝑃𝑅
……………(1)
∵Corresponding angles are also equal in two similar triangles
∴∠A = ∠P, ∠B = ∠Q and ∠C = ∠R ...(2)
Since AD and PM are medians
∴BC = 2 BD and QR = 2 QM
∴From (1),
