Advertisements
Advertisements
प्रश्न
If the perpendicular bisector of the sides of a triangle PQR meet at I, then prove that the line joining from P, Q, R to I are equal.
Advertisements
उत्तर
Given:
In ΔPQR,
PA is the perpendicular bisector of QR ⇒ QA = RA
RC is the perpendicular bisector of PQ ⇒ PC = QC
QB is the perpendicular bisector of PR ⇒ PR = RB
PA, RC and QB meet at I.
To prove: IP = IQ = IR
Proof:
In ΔQIA and ΔRIA
QA = RA ....[Given]
∠QAI = ∠RAI ....[Each = 90]
IA = IA ....[Common]
∴ By Side-Angle-Side criterion of congruence,
ΔIQ = IR ....(i)
Similarly, in ΔRIB and ΔPIB
RB = PB ...[Given]
∠RBI = ∠PBI ...[Each = 90°]
IB = IB ...[Common]
∴ By Side-Angle-Side criterion of congruence,
ΔRIB ≅ ΔPIB
The corresponding parts of the congruent triangles are congruent.
∴ IR = IP ....(ii)
From (i) and (ii), we have
IP = IQ = IR.
APPEARS IN
संबंधित प्रश्न
If ΔDEF ≅ ΔBCA, write the part(s) of ΔBCA that correspond to `bar(DF)`
In the following example, a pair of triangles is shown. Equal parts of triangle in each pair are marked with the same sign. Observe the figure and state the test by which the triangle in each pair are congruent.
By ______ test
ΔXYZ ≅ ΔLMN
Observe the information shown in pair of triangle given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.
From the information shown in the figure,
In ΔABC and ΔPQR
∠ABC ≅ ∠PQR
seg BC ≅ seg QR
∠ACB ≅ ∠PRQ
∴ ΔABC ≅ ΔPQR ...`square` test
∴ ∠BAC ≅ `square` ...corresponding angles of congruent triangles.
`{:("seg AB" ≅ square),("and" square ≅ "seg PR"):}}` ...corresponding sides of congruent triangles
In the following figure, OA = OC and AB = BC.
Prove that:
(i) ∠AOB = 90o
(ii) ΔAOD ≅ ΔCOD
(iii) AD = CD
The following figure has shown a triangle ABC in which AB = AC. M is a point on AB and N is a point on AC such that BM = CN.
Prove that: (i) AM = AN (ii) ΔAMC ≅ ΔANB
State, whether the pairs of triangles given in the following figures are congruent or not:
Δ ABC in which AB = 2 cm, BC = 3.5 cm and ∠C = 80° and Δ DEF in which DE = 2 cm, DF = 3.5 cm and ∠D = 80°.
A is any point in the angle PQR such that the perpendiculars drawn from A on PQ and QR are equal. Prove that ∠AQP = ∠AQR.
AD and BE are altitudes of an isosceles triangle ABC with AC = BC. Prove that AE = BD.
In the given figure ABCD is a parallelogram, AB is Produced to L and E is a midpoint of BC. Show that:
a. DDCE ≅ DLDE
b. AB = BL
c. DC = `"AL"/(2)`
“If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the two triangles must be congruent.” Is the statement true? Why?
