In one of the residential colony of city, there is a triangular park ABC available. The Resident
Welfare Association of the colony wishes to divide this park into two parts of equal areas. One for
planting trees and raising a lawn and the other for providing place for children park for playing
activities. One of the members Ram suggested to draw a line segment XY || BC for the same.
(i) Which is the correct similarity criteria is used for above triangles ABC and XBY?
(a) SSS (b) AAA (c) RHS (d) ASA
(ii) What is the ratio of areas of ∆ABC : ∆ XBY?
(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 1 : √2
(iii) If ∠BXY = 60°, then ∠BAC is :
(a) 80° (b) 60° (c) 140° (d) 100°
(iv) If the ratio of areas of ∆BXY : ∆ ABC is 1 : 2, then XB : AB is :
(a) 1 : 2 (b) 1 : 4 (c) 2 : 4 (d) 1 : √2
(v) If the area of ∆ AXY is 128 m2
, then what will be a
Answers
(i) Which is the correct similarity criteria is used for above triangles ABC and XBY ?
(a) SSS (b) AAA (c) RHS (d) ASA
Answer :-
In ∆ABC and ∆XBY we have,
→ ∠ABC = ∠XBY (common)
→ ∠BAC = ∠BXY (corresponding angles.)
→ ∠BCA = ∠BYX (corresponding angles.)
so,
→ ∆ABC ~ ∆XBY (By AAA similarity.) (b)
(ii) What is the ratio of areas of ∆ABC : ∆ XBY ?
(a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 1 : √2
Answer :-
given that,
→ Area ∆(BXY) = Area quad.(XYCA)
so,
→ Area ∆ABC : Area ∆XBY = (Area ∆(BXY) + Area quad.(XYCA)) : Area ∆XBY = Area ∆XBY + Area ∆XBY : Area ∆XBY = 2 * Area ∆XBY : Area ∆XBY = 2 : 1 (b) .
(iii) If ∠BXY = 60°, then ∠BAC is :
(a) 80° (b) 60° (c) 140° (d) 100°
Answer :-
→ ∠BAC = ∠BXY (corresponding angles.)
→ ∠BAC = 60° (b)
(iv) If the ratio of areas of ∆BXY : ∆ ABC is 1 : 2, then XB : AB is :
(a) 1 : 2 (b) 1 : 4 (c) 2 : 4 (d) 1 : √2
Answer :-
since ∆BXY ~ ∆ABC
so,
→ Area ∆BXY : Area ∆ABC = XB² : AB²
→ 1 : 2 = XB² : AB²
→ (1/2) = (XB/AB)²
→ 1/√2 = XB/AB
→ XB : AB = 1 : √2 (d)
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