Maths-
General
Easy

Question

The Quadrilateral PQRS has angles at  S,Q right angles and the diagonals PR, QS are perpendicular. Prove that SR = QR.

hintHint:

Hint :- We know that if a quadrilateral has a sum of opposite angles 180° then it is a cyclic quadrilateral . Now , find the relation between OS and OQ. By proving congruence of triangles, prove SR= QR .

The correct answer is: Hence proved


    Solution :-
    Aim  :- Prove that SR = QR
    Explanation(proof ) :-
    We know that if a quadrilateral has a sum of opposite angles 180°
    ∠S  + ∠ Q = 90 + 90 = 180°(S and Q are opposite angle)
    Here PQRS is cyclic and ∠S = 90°
    As the angle angle in a semicircle is  90° then PSR lies in a semi circle
    and PR is diameter.
    As PR passes through centre and perpendicular to chord SQ
    The  perpendicular drawn from centre to chord bisects the chord
    So, OS = OQ as PR bisects chord SQ
    In  straight capital delta OQR and  straight capital delta OSR
    OS = OQ (side)
    ∠SOR = ∠QOR =90° (Angle)
    OP =OP (common side )
    By SAS rule straight capital delta OQR ≅  straight capital delta OSR
    By congruence we get , SR= QR
    Hence proved

    Related Questions to study

    General
    Maths-

    P and Q are points on the sides CA and CB  respectively of a straight triangle ABC  right angled at  c. Prove that AQ squared plus BP squared equalsA B squared plus P Q squared

    P and Q are points on the sides CA and CB  respectively of a straight triangle ABC  right angled at  c. Prove that AQ squared plus BP squared equalsA B squared plus P Q squared

    Maths-General
    General
    Maths-

    In △ABC, ∠B=90 and  is the mid point of BC. Prove that A C squared minus A D squared equals 3 B D squared
     

    In △ABC, ∠B=90 and  is the mid point of BC. Prove that A C squared minus A D squared equals 3 B D squared
     

    Maths-General
    General
    Maths-

    ABC text  is an equilateral triangle  end text. straight K text  is a point on  end text BC text  such that  end text BK colon KC equals 2 colon 1 text . Prove that  end text 9 AK squared equals 7 AC squared

    ABC text  is an equilateral triangle  end text. straight K text  is a point on  end text BC text  such that  end text BK colon KC equals 2 colon 1 text . Prove that  end text 9 AK squared equals 7 AC squared

    Maths-General
    parallel
    General
    Maths-

    The perpendicular AD on the base BC of Triangle ABC intersects BC in D such that BD = 3CD. Prove that
    2 A B squared equals 2 A C squared plus B C squared

    The perpendicular AD on the base BC of Triangle ABC intersects BC in D such that BD = 3CD. Prove that
    2 A B squared equals 2 A C squared plus B C squared

    Maths-General
    General
    Maths-

    In an equilateral straight triangle ABC text ,  end textthe side BC  is trisected at D . Prove that 9 A D squared equals 7 A B squared
    .

    In an equilateral straight triangle ABC text ,  end textthe side BC  is trisected at D . Prove that 9 A D squared equals 7 A B squared
    .

    Maths-General
    General
    Maths-


    ABC and DBC are two isosceles triangles on the same base BC. Show
    that straight angle ABD equals straight angle ACD


    ABC and DBC are two isosceles triangles on the same base BC. Show
    that straight angle ABD equals straight angle ACD

    Maths-General
    parallel
    General
    Maths-

    PQ and RS are respectively the smallest and longest sides of a quadrilateral PQRS. Show that ∠ P > ∠ R

    PQ and RS are respectively the smallest and longest sides of a quadrilateral PQRS. Show that ∠ P > ∠ R

    Maths-General
    General
    Maths-

    Classify whether the following Surd is rational or irrational , justify. 3√24− 5√5
     

    Classify whether the following Surd is rational or irrational , justify. 3√24− 5√5
     

    Maths-General
    General
    Maths-

    In △ABC, AD and  are the bisectors of ∠A and ∠C respectively. AD and CE intersect at . If ∠ABC = 90^∘, then find ∠AOC.

    In △ABC, AD and  are the bisectors of ∠A and ∠C respectively. AD and CE intersect at . If ∠ABC = 90^∘, then find ∠AOC.

    Maths-General
    parallel
    General
    Maths-

    A B C text  is a right triangle with  end text straight angle A equals 90 to the power of ring operator BL text  and  end text CM text  are its medians. Prove that  end text 4 open parentheses BL squared plus CM squared close parentheses equals 5 BC squared text .  end text

    A B C text  is a right triangle with  end text straight angle A equals 90 to the power of ring operator BL text  and  end text CM text  are its medians. Prove that  end text 4 open parentheses BL squared plus CM squared close parentheses equals 5 BC squared text .  end text

    Maths-General
    General
    Maths-

    Classify whether the following Surd is rational or irrational , justify 4√5+ 9

    Classify whether the following Surd is rational or irrational , justify 4√5+ 9

    Maths-General
    General
    Maths-

    PQR is a triangle in which PQ=PR.S is any point on the side PQ. Through S, a line is drawn parallel to QR intersecting PR at T. Prove that PS = PT.

    PQR is a triangle in which PQ=PR.S is any point on the side PQ. Through S, a line is drawn parallel to QR intersecting PR at T. Prove that PS = PT.

    Maths-General
    parallel
    General
    Maths-

    text  In Triangle  end text A B C text , If  end text A D text  is the median show that,  end text A B squared plus A C squared equals 2 open parentheses A D squared plus B D squared close parentheses

    text  In Triangle  end text A B C text , If  end text A D text  is the median show that,  end text A B squared plus A C squared equals 2 open parentheses A D squared plus B D squared close parentheses

    Maths-General
    General
    Maths-

    Classify whether the following Surd is rational or irrational , justify 8-√48

    Classify whether the following Surd is rational or irrational , justify 8-√48

    Maths-General
    General
    Maths-

    Classify whether the following Surd is rational or irrational , Justify.√47+√8

    Classify whether the following Surd is rational or irrational , Justify.√47+√8

    Maths-General
    parallel

    card img

    With Turito Academy.

    card img

    With Turito Foundation.

    card img

    Get an Expert Advice From Turito.

    Turito Academy

    card img

    With Turito Academy.

    Test Prep

    card img

    With Turito Foundation.