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Triangles

Short Answer Type

501. In figure, AB > AC, BO and CO are the bisectors of ∠B and ∠C respectively. Show that OB > OC.

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502.

In the given figure, ∠ABD = 130° and ∠EAC = 120°. Prove that AB > AC.

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503. If AD is the bisector of ∠A of ∆ ABC, show that AB > DB.

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Long Answer Type

504. 0 is a point in the interior of ∆PQR.

OP + OQ + OR > $1 half$ (PQ + QR + RP).

Given: O is a point in the interior of ∆PQR.

To Prove: OP + OQ + OR >

1 half

(PQ + QR + RP)
In ∆OPQ,
OP + OQ > PQ ...(1)

Given: O is a point in the interior of ∆PQR.To Prove: OP + OQ + OR

|∵ The sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆OQR,
OQ + OR > QR ...(2)
| ∵ The sum of the lengths of any two sides of a triangle is greater than the length of the third
side
In AORP,
OR + OP > RP ...(3)
| ∵ The sum of the lengths of any two sides of a triangle is greater than the length of the third side
From (1), (2) and (3), we get,
(OP + OQ) + (OQ + OR) + (OR + OP) > (PQ + QR + RP)
⇒ 2(OP + OQ + OR) > (PQ + QR + RP)

$rightwards double arrow space space space OP plus OQ plus OR greater than 1 half left parenthesis PQ plus QR plus RP right parenthesis$

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Short Answer Type

505. In figure, D is any point on the base BC produced of an isosceles triangle ∆ABC. Prove that AD > AB.

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506. In figure, S is any point in its interior of ∆PQR. Show that SQ + SR < PQ + PR.

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507. Prove that the difference of any two sides of a triangle is less than the third side.

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Multiple Choice Questions

508. In the following figure, ∠B > ∠A and ∠D > ∠E. Then, the relation between AE and BD is

AE = BD
AE > BD
AE < BD
AE < BD

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509. In ∆ABC, AB > AC and D is any point on side BC. Then, the relation between AB and AD is

AB > AD
AB = AD
AB < AD
AB < AD

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510. In the following figure, in ∆ABC, AB = AC; CD = CA and ∠ADC = 20°. Then, ∠ABC =

10°
20°
30°
30°

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

Long Answer Type

504. 0 is a point in the interior of ∆PQR.OP + OQ + OR > (PQ + QR + RP).

Short Answer Type

Multiple Choice Questions

504. 0 is a point in the interior of ∆PQR.

OP + OQ + OR > $1 half$ (PQ + QR + RP).