### Circles Class 9th Mathematics AP Board Solution

##### Question 1.Find the values of x and y in the figures given below.i. ii. iii. Answer:(i) We know “Sum of all angles of a triangle is 180 ”. x + y + 30 = 180 Since it is an isosceles triangle, x = y. x + x + 30 = 180 ⇒ 2x + 30 = 180 ⇒ 2x = 180 -30 ⇒ 2x = 150 ⇒ x = ⇒ x = 75  x = y = 75 (ii) We know that “Angles on the opposite sides in the cyclic quadrilateral are supplementary”. x + 110 = 180 ⇒ x = 180 -110 ⇒ x = 70  y + 85 = 180 ⇒ y = 180 -85 ⇒ y = 95 (iii) Given, x = 90° x + y + 50 = 180 ⇒ 90 + y + 50 = 180 ⇒ y + 140 = 180 ⇒ y = 180 -140 ⇒ y = 40 Question 2.Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also ∠A + ∠C = 180°, then prove that the vertex D also lie on the same circle.Answer:In a quadrilateral, if the sum of opposite angles is 180°, then it is acyclic quadrilateral.In quad. ABCD, A + C = 180°,Therefore, ABCD is a cyclic quad.In a cyclic quad., the vertices lie on the same circle. Thus, D alsolies on the same circle.Hence, proved.Question 3.Prove that a cyclic rhombus is a square.Answer:To prove rhombus inscribed in a circle is a square, we need to prove that either any one of its interior angles is equal to 90° or its diagonals are equal. ∠ABD = ∠DBC = b∠ADB = ∠BDC = aIn the figure, diagonal BD is angular bisector of angle B and angle D.In triangle ABD and BCD,AD = BC (sides of rhombus are equal)AB = CD (sides of rhombus are equal)BD = BD (common side)△ABD ≅ △BCD. (SSS congruency)In the figure,2a + 2b = 180° (as, opposite angles of a cyclic quadrilateral are always supplementary)2(a + b) = 180°a + b = 90°In △ABD,Angle A = 180°-(a + b)= 180°-90°= 90°Therefore, proved that one of its interior angle is 90°Hence, rhombus inscribed in a circle is a square.Question 4.For each of the following, draw a circle and inscribe the figure given. If a polygon of the given type can’t be inscribed, write not possible.(a) Rectangle(b) Trapezium(c) Obtuse triangle(d) Non-rectangular parallelogram(e) Accute issosceles triangle(f) A quadrilateral PQRS with as diameter.Answer:(a) Rectangle = Possible (b) Trapezium = Possible (c) Obtuse triangle = Not Possible (d) Non-rectangular parallelogram = Not Possible (e) Acute isosceles triangle = Possible (f) A quadrilateral PQRS with as diameter = Possible PDF FILE TO YOUR EMAIL IMMEDIATELY PURCHASE NOTES & PAPER SOLUTION. @ Rs. 50/- each (GST extra)

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# 2019 Board Paper Solution

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