If chords AB and CD of a circle intersect each other at a point P inside the circle, prove that : PA × PB = PC × PD.

In △PAC and △PDB ∠APC = ∠DPB (Vertically opposite angles are equal) ∠CAP = ∠BDP (Angles in same segment are equal) ∴ △PAC ~ △PDB We know that, When two triangles are similar, the ratio of the lengths of their corresponding sides are proportional. ∴ ⇒ PA × PB = PC × PD. Hence, proved that PA × PB = PC × PD.

The line 4x + 5y + 20 = 0 meets x-axis at point A and y-axis at point B. Find :
(i) the coordinates of points A and B.
(ii) the co-ordinates of point P in AB such that AB : BP = 5 : 3.
(iii) the equation of the line through P and perpendicular to AB.

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Use ruler and a pair of compasses only in this question :
(i) Draw a circle on AB = 6.4 cm as diameter.
(ii) Construct another circle of radius 2.5 cm to touch the circle in (i) above and the diameter AB produced.

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P and Q are centers of a circle of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cm which touches the above circles externally. Given that ∠PRQ = 90°, write an equation in x and solve it.

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In the given figure, DE || BC and AE : EC = 5 : 4. Find :
(i) DE : BC
(ii) DO : DC
(iii) area of △DOE : area of △ DCE.

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Mathematics

If chords AB and CD of a circle intersect each other at a point P inside the circle, prove that : PA × PB = PC × PD.

Circles

Related Questions

The line 4x + 5y + 20 = 0 meets x-axis at point A and y-axis at point B. Find :
(i) the coordinates of points A and B.
(ii) the co-ordinates of point P in AB such that AB : BP = 5 : 3.
(iii) the equation of the line through P and perpendicular to AB.

Use ruler and a pair of compasses only in this question :
(i) Draw a circle on AB = 6.4 cm as diameter.
(ii) Construct another circle of radius 2.5 cm to touch the circle in (i) above and the diameter AB produced.

P and Q are centers of a circle of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cm which touches the above circles externally. Given that ∠PRQ = 90°, write an equation in x and solve it.

In the given figure, DE || BC and AE : EC = 5 : 4. Find :
(i) DE : BC
(ii) DO : DC
(iii) area of △DOE : area of △ DCE.

Mathematics

If chords AB and CD of a circle intersect each other at a point P inside the circle, prove that : PA × PB = PC × PD.

Circles

Related Questions

The line 4x + 5y + 20 = 0 meets x-axis at point A and y-axis at point B. Find :(i) the coordinates of points A and B.(ii) the co-ordinates of point P in AB such that AB : BP = 5 : 3.(iii) the equation of the line through P and perpendicular to AB.

Use ruler and a pair of compasses only in this question :(i) Draw a circle on AB = 6.4 cm as diameter.(ii) Construct another circle of radius 2.5 cm to touch the circle in (i) above and the diameter AB produced.

P and Q are centers of a circle of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cm which touches the above circles externally. Given that ∠PRQ = 90°, write an equation in x and solve it.

In the given figure, DE || BC and AE : EC = 5 : 4. Find :(i) DE : BC(ii) DO : DC(iii) area of △DOE : area of △ DCE.

The line 4x + 5y + 20 = 0 meets x-axis at point A and y-axis at point B. Find :
(i) the coordinates of points A and B.
(ii) the co-ordinates of point P in AB such that AB : BP = 5 : 3.
(iii) the equation of the line through P and perpendicular to AB.

Use ruler and a pair of compasses only in this question :
(i) Draw a circle on AB = 6.4 cm as diameter.
(ii) Construct another circle of radius 2.5 cm to touch the circle in (i) above and the diameter AB produced.

In the given figure, DE || BC and AE : EC = 5 : 4. Find :
(i) DE : BC
(ii) DO : DC
(iii) area of △DOE : area of △ DCE.