In the given triangle PQR, AB || QR, QP || CB and AR intersects CB at O. Using the given diagram answer the following question: (i) The triangle similar to ΔARQ is (a) ΔORC (b) ΔARP (c) ΔOBR (d) ΔQRP (ii) ΔPQR ~ΔBCR by axiom (a) SAS (b)AAA (c) SSS (d) AAS (iii) If QC =6 cm, CR = 4 cm, BR = 3 cm. The length of RP is (a) 4.5 cm (b) 8cm (c) 7.5cm (d) 5cm (iv) The ratio PQ: BC is (a) 2 : 3 (b) 3 : 2 (c) 5 : 2 (d) 2 : 5
Answers
Answer:
(i) a) ΔORC
(ii) b) AAA
(iii) c) 7.5 cm
iv) c) 5:2
Step-by Step explanation
iii) Since ΔPQR ~ΔBCR
PQ/BC = QR/CR = PR/BR
QR=QC+CR
QR=6+4
QR=10cm
Therefore,
QR/CR = PR/BR
10/4 = PR/3
5/2 *3 = PR
PR= 15/2 = 7.5 cm
iv) PQ/BC = QR/CR
PQ/BC = 10/4
PQ/BC = 5/2
PQ:BC = 5:2
Given :- In the given triangle PQR, AB || QR, QP || CB and AR intersects CB at O.
Solution :-
(i)
In ∆ARQ and ∆ORC we have,
→ AQ || OC { Since we have given that, QP || CB }
So,
→ ∠QAR = ∠COR { Corresponding angles }
→ ∠ARQ = ∠ORC { Common angles }
→ ∠AQR = ∠OCR { Corresponding angles }
then,
→ ∆ARQ ~ ∆ORC { By AAA similarity }
Therefore, Option (A) ∆ORC is correct answer .
(ii)
In ∆PQR and ∆BCR we have,
→ QP || CB { given }
So,
→ ∠QPR = ∠CBR { Corresponding angles }
→ ∠PRQ = ∠BRC { Common angles }
→ ∠PQR = ∠BCR { Corresponding angles }
then,
→ ∆PQR ~ ∆BCR { By AAA similarity }
Therefore, Option (b) AAA is correct answer .
(iii)
from (ii) we get,
→ ∆PQR ~ ∆BCR
So,
→ PQ/BC = QR/CR = PR/BR { when two ∆'s are similar their corresponding sides are in same ratio.}
→ QR/CR = PR/BR
→ (QC + CR)/CR = (PB + BR)/BR ------------ Equation (1)
given that, QC =6 cm, CR = 4 cm, BR = 3 cm.
Let PB = x cm . Putting all these values in Eqn.(1) now,
→ (6 + 4)/4 = (x + 3)/3
→ 10/4 = (x + 3)/3
→ 5/2 = (x + 3)/3
→ 15 = 2(x + 3)
→ 15 = 2x + 6
→ 15 - 6 = 2x
→ 2x = 9
→ x = 4.5
therefore,
→ RP = RB + BP
→ PR = BR + PB
→ PR = 3 + 4.5
→ PR = 7.5 cm .
Therefore, Option (c) 7.5 cm is correct answer .
(iv)
from (ii) again,
→ ∆PQR ~ ∆BCR
So,
→ PQ/BC = QR/CR = PR/BR { when two ∆'s are similar their corresponding sides are in same ratio.}
→ PQ/BC = PR/BR
→ PQ/BC = 7.5/3
→ PQ/BC = 75/30
→ PQ/BC = 5/2
→ PQ : BC = 5 : 2
Therefore, Option (c) 5 : 2 is correct answer .
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