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Ask QuestionPosted by Mohak Dubey 2 years, 4 months ago
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Preeti Dabral 2 years, 4 months ago
Given: ABCD is a square, where ∠PQR = 90o and PB = QC = DR
To prove: (i) QB = RC (ii) PQ = QR
(iii) ∠QPR = 45o
Proof:
- Here,
BC = CD … (Sides of square)
CQ = DR … (Given)
BC = BQ + CQ
∴ CQ = BC − BQ
∴ DR = BC – BQ .......(1)
Also,
CD = RC + DR
∴ DR = CD - RC = BC - RC ........(2)
From (1) and (2), we have,
BC - BQ = BC - RC
∴ BQ = RC - Now in △RCQ and △QBP
we have, PB = QC … (Given)
BQ = RC … [from (i)]
∠RCQ = ∠QBP … 90o each
Hence by SAS(Side-Angle-Side) congruence rule,
△RCQ ≅ △QBP
∴ QR = PQ … (by cpct) - △RCQ ≅ △QBP and QR = PQ … [from (2)]
∴ In △RPQ,
∠QPR = ∠QRP =12(180o - 90o) = 902= 45o
∴ ∠QPR = 45o
Posted by Hena Kausar 2 years, 4 months ago
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Posted by Jinali Zaveri 2 years, 4 months ago
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Preeti Dabral 2 years, 4 months ago
Given: In right triangle ABC, right angled at C. M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B.
To Prove:
- △AMC ≅ △BMD
- ∠DBC is a right angle
- △DBC ≅ △ACB
- CM = 12AB
Proof:
- In △AMC and △BMD
AM = BM ...[As M is the mid-point]
CM = DM ...[Given]
∠AMC = ∠BMD ...[Vertically opposite angles]
∴ △AMC ≅ △BMD proved ...[SAS property] ...(1) - △AMC ≅ △BMD ...[From (1)]
∠ACM = ∠BDM ...[c.p.c.t.]
These are alternate interior angles and they are equal.
∴ AC ‖ BD
As AC ‖ BD and transversal BC intersects them
∴ ∠DBC + ∠ACB = 180° ...[Sum of the consecutive interior angles of the transversal]
∠DBC + 90° = 180°
∠DBC = 180° - 90° = 90°
∠DBC is a right angle proved. - △AMC ≅ △BMD ...[From (1)]
∴ AC = BD ...[c.p.c.t.] ...(2)
In DDBC and DACB
BC = CB ...[Common]
∠DBC = ∠ACB ...[each = 90° as proved above]
BD = CA ...[From (2)]
∴ △DBC ≅ △ACB ...[SAS property] - DDBC ≅ DACB ...[As proved in (iii)]
∴ DC = AB ...[c.p.c.t.]
∴ 2CM = AB ...[DM = CM = 12 DC]
∴ CM =12 AB
Posted by Sonali Verma 2 years, 4 months ago
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Preeti Dabral 2 years, 4 months ago
Let a square ABCD in which L, M, N & O are the midpoints.
In △AML and △CNO
AM = CO (AB = DC and M and O are the midpoints)
AL = CN (AD = BC and L and N are the midpoints)
∠ MAL = ∠NCO (all angles of a square = 90°)
by AAS criteria
△AML ≅ △CNO
∴ ML = ON (CPCT)
similarly △MBN ≅ △LDO
now,
in △AML,
∠AML = ∠ALM (AM = AL)
= 45°
similarly in △LDO
∠DLO = 45°
∴ ∠ MLO = 90°
by the properties of SQUARE
All sides are equal and angles are 90°
Posted by Harshdeep Raj 2 years, 4 months ago
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Posted by Allpesha Shambharkar 2 years, 4 months ago
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Preeti Dabral 2 years, 4 months ago
Factor Theorem is generally applied to factoring and finding the roots of polynomial equations. It is the reverse form of the remainder theorem. Problems are solved based on the application of synthetic division and then to check for a zero remainder.
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Disha Kashyap 😊 2 years, 4 months ago
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