Two equal circle touch externally at point P. PX and PY are chords in the respective circles which are perpendicular. If the centres Are A and B. Show that AX is parallel to BY and AXYB is a parallelogram.
Draw the diameters through A and B and P. The Draw lines connecting X and Y to the ends opposite P. Now you have two congruent right triangles. The rest follows pretty easily.
To show that AX is parallel to BY and that AXYB is a parallelogram, we can use some basic geometry concepts and properties.
Let's start by drawing a diagram to visualize the given information. We have two circles with centers A and B, respectively. The circles touch externally at point P. Let PX and PY be chords in their respective circles, which are perpendicular.
To show that AX is parallel to BY and that AXYB is a parallelogram, we need to prove two things:
1. AX is parallel to BY.
2. The opposite sides of the quadrilateral AXYB are equal in length.
Let's tackle these one by one.
1. To show that AX is parallel to BY, we can use the property of tangents drawn from an external point to a circle. Since PX and PY are chords of their respective circles, they must pass through the centers A and B, respectively. Hence, we can draw the radii AP and BP, which are both perpendicular to PX and PY, respectively.
Now, since PX and PY are perpendicular to the radii AP and BP, we can conclude that angles APX and BPY are both right angles. From this, we can apply the property of opposite angles formed by intersecting lines, which states that if two lines intersect and one pair of opposite angles is equal, then the lines are parallel. Therefore, we can say that AX is parallel to BY.
2. To show that the opposite sides of the quadrilateral AXYB are equal in length, we can use the property of congruent chords in a circle. Since the two circles are equal, the lengths of the chords PX and PY are equal. Additionally, since the circle with center A touches point P, the length of AX will be half the length of PX. Similarly, since the circle with center B touches point P, the length of BY will be half the length of PY.
From this information, we can conclude that AX = PX/2 and BY = PY/2. Since PX = PY, we have AX = BY.
Therefore, we have shown that AX is parallel to BY, and the opposite sides of the quadrilateral AXYB are equal in length. Hence, AXYB is a parallelogram.
In summary: To show that AX is parallel to BY and that AXYB is a parallelogram, we used the properties of perpendicular chords, tangents, congruent chords, and opposite angles formed by intersecting lines.