Advertisements
Advertisements
Question
The given figure shows a circle with center O. P is mid-point of chord AB.
Show that OP is perpendicular to AB.
Advertisements
Solution
Given: in the figure, O is center of the circle, and AB is chord. P is a point on AB such that AP = PB.
We need to prove that, OP ⊥ AB
Construction: Join OA and OB
Proof:
In ΔOAP and ΔOBP
OA = OB ...[Radii of the same circle]
OP = OP ...[Common]
AP = PB ...[Given]
∴ By Side-Side-Side criterion of congruency,
ΔOAP ≅ ΔOBP
The corresponding parts of the congruent triangles are congruent.
∴ ∠OPA = ∠OPB ...[By c.p.c.t]
But ∠OPA + ∠OPB = 180° ...[Linear pair]
∴ ∠OPA = ∠OPB = 90°
Hence, OP ⊥ AB.
APPEARS IN
RELATED QUESTIONS
AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠BAD = ∠ABE and ∠EPA = ∠DPB (See the given figure). Show that
- ΔDAP ≅ ΔEBP
- AD = BE
You want to show that ΔART ≅ ΔPEN,
If you have to use SSS criterion, then you need to show
1) AR =
2) RT =
3) AT =
In ΔABC, ∠A = 30°, ∠B = 40° and ∠C = 110°
In ΔPQR, ∠P = 30°, ∠Q = 40° and ∠R = 110°
A student says that ΔABC ≅ ΔPQR by AAA congruence criterion. Is he justified? Why or why not?
In the figure, the two triangles are congruent.
The corresponding parts are marked. We can write ΔRAT ≅ ?
Which of the following statements are true (T) and which are false (F):
Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal equal to the hypotenuse and a side of the other triangle.
In the given figure, prove that:
CD + DA + AB + BC > 2AC
In the given figure, prove that:
CD + DA + AB > BC
ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.
If the following pair of the triangle is congruent? state the condition of congruency:
In ΔABC and ΔPQR, AB = PQ, AC = PR, and BC = QR.
In a ΔABC, BD is the median to the side AC, BD is produced to E such that BD = DE.
Prove that: AE is parallel to BC.
