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Ch 6 Ex 6.3 Q12 to Q16

Rashmi Taneja
Rashmi Taneja

1) Two triangles ABC and PQR are similar if their corresponding sides and medians are proportional. Specifically, if sides AB and BC are proportional to sides PQ and QR, and medians AD and PM are also proportional, then the two triangles are similar. 2) If an angle of one triangle is equal to a corresponding angle of another triangle, and the ratios of their sides are also equal, then the triangles are similar by the AA and SAS similarity criteria. 3) The height of a tower can be calculated using similar right triangles if the length of a pole, its shadow, and the shadow of the tower are known, since the angles of elevation of the sun are the same.

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CH 6 TRIANGLES EX 6.3 Q12 TO Q 16 PREPARED BY RASHMI TANEJA IPS
12. Sides AB and BC and median AD of triangle ABC are respectively proportional to sides PQ and QR and median PM of PQR (see figure). Show that ABC ~ PQR. Sol. We have ABC and PQR in which AD and PM are medians are propotional to corresponding sides BC and QR respectively such, that 基 = 巨 = 基 基 = 1 2 巨 1 2 = 基 基 = 巨 = 基 Using SSS similarity, we have: Their corresponding q es are equal ABD = PQM 癌ABC = PQR Now, in ABC and PQR 基 = 巨 (i) Given ABC = PQR ..(ii) proved above ABC ~ PQR by SAS similarity criteria
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 ABC and point D on its side BC such that ADC = BAC In ABC and ADC BAC = ADC [Given] And BCA = DCA Using AA similarity, we have BAC ~ ADC Their corresponding sides are proportional
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 ABC ~ PQR. Sol. We have two s ABC and PQR such that AD and PM are medians proportional to corresponding sides BC and QR respectively. Also 基 = 巨 = 基 基 = 1 2 巨 1 2 = 基 基 = 巨 = 基 since, the corresponding angles of similar triangles are equal. 癌ABD = PQM ABC = PQR ...(2) Now, in ABC and PQR ABC = PQR [From (2)] 基 = 巨 By SAS ABC ~ PQR.
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 ABC), 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 ABC and DEF we have B = E = 90属 A = D [Angular elevation of the sun at the same time]. Using AA criteria of similarity, we have ABC ~ DEF Their sides are proportional Thus, the required height of the tower is 42 m.
16. If AD and PM are medians of triangles ABC and PQR, respectively where ABC ~ PQR, prove that 基 = 基 Sol. We have ABC ~ PQR such that AD and PM are the medians. 砧ABC ~ PQR And the corresponding sides of similar triangles are proportional. 基 = 巨 = 基 (1) Corresponding angles are also equal in two similar triangles 癌A = P, B = Q and C = R ...(2) Since AD and PM are medians BC = 2 BD and QR = 2 QM From (1),

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Ch 6 Ex 6.3 Q12 to Q16

  • 1. CH 6 TRIANGLES EX 6.3 Q12 TO Q 16 PREPARED BY RASHMI TANEJA IPS
  • 2. 12. Sides AB and BC and median AD of triangle ABC are respectively proportional to sides PQ and QR and median PM of PQR (see figure). Show that ABC ~ PQR. Sol. We have ABC and PQR in which AD and PM are medians are propotional to corresponding sides BC and QR respectively such, that 基 = 巨 = 基 基 = 1 2 巨 1 2 = 基 基 = 巨 = 基 Using SSS similarity, we have: Their corresponding q es are equal ABD = PQM 癌ABC = PQR Now, in ABC and PQR 基 = 巨 (i) Given ABC = PQR ..(ii) proved above ABC ~ PQR 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 ABC and point D on its side BC such that ADC = BAC In ABC and ADC BAC = ADC [Given] And BCA = DCA Using AA similarity, we have BAC ~ 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 ABC ~ PQR. Sol. We have two s ABC and PQR such that AD and PM are medians proportional to corresponding sides BC and QR respectively. Also 基 = 巨 = 基 基 = 1 2 巨 1 2 = 基 基 = 巨 = 基 since, the corresponding angles of similar triangles are equal. 癌ABD = PQM ABC = PQR ...(2) Now, in ABC and PQR ABC = PQR [From (2)] 基 = 巨 By SAS ABC ~ PQR.
  • 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 ABC), 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 ABC and DEF we have B = E = 90属 A = D [Angular elevation of the sun at the same time]. Using AA criteria of similarity, we have ABC ~ DEF 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 ABC ~ PQR, prove that 基 = 基 Sol. We have ABC ~ PQR such that AD and PM are the medians. 砧ABC ~ PQR And the corresponding sides of similar triangles are proportional. 基 = 巨 = 基 (1) Corresponding angles are also equal in two similar triangles 癌A = P, B = Q and C = R ...(2) Since AD and PM are medians BC = 2 BD and QR = 2 QM From (1),