Question

construct a tangent to the circle at point a.

1. construct a diameter through points p and a.
2. construct a perpendicular to line pa at point a.

hint

Explanation:

Step1: Understand the circle and diameter

We have a circle with center (implied by diameter PA, so P is the center? Wait, no, diameter is a line through the center, so if PA is a diameter, then P is the center. So the circle has center P, and A is on the circle.

Step2: Recall tangent properties

A tangent to a circle at a point is perpendicular to the radius (or diameter, since diameter is a line through center, so the radius is PA here) at that point. So to construct the tangent at A, we need a line perpendicular to PA at A.

Step3: Construct the perpendicular

Using a compass:
- With A as the center, draw an arc that intersects PA at two points (let's say X and Y, on either side of A, but since PA is a diameter, maybe we can adjust. Wait, the first step is to have diameter PA. Then, to construct the perpendicular at A to PA:
- Place the compass point at A, and draw an arc with radius more than half of the distance from A to P (but actually, for perpendicular to a line at a point:
- Open the compass to a suitable radius, place the point at A, draw two arcs intersecting PA at two points, say, left and right of A (but since PA is a straight line, let's say we draw an arc above and below the line PA, intersecting PA at two points, say, M and N (on PA, on either side of A? Wait, no, A is on PA, so if we center at A, and draw an arc with radius r, it will intersect PA at A - r and A + r (along the line PA). Then, from those two intersection points (let's call them C and D, where C is on PA towards P, and D is on PA away from P? No, PA is a diameter, so P is the center, so A is on the circle, PA is the radius? Wait, no, diameter is twice the radius. Wait, maybe P is the center, so PA is the radius? Wait, the problem says "construct a diameter through points P and A", so that means PA is a diameter, so the length from P to A is the diameter, so the center is the midpoint of PA. Wait, maybe I misread. Let's re-express:

To construct the tangent at A:

1. Since PA is a diameter (so the line through P and A passes through the center, so the center is the midpoint of PA).
2. The tangent at A is perpendicular to the radius (or diameter, since diameter is a line through center, so the radius is from center to A, which is half of PA). So the tangent line at A must be perpendicular to PA at A.

To construct the perpendicular:

- Using a compass, with A as the center, draw an arc that intersects PA at two points (let's say, B and C, on PA, on either side of A, but since A is on PA, we can draw an arc with radius, say, 2 units, so it intersects PA at A - 2 and A + 2 (along the line PA). Then, from each of those two points (B and C), draw arcs above and below the line PA, with the same radius (larger than the distance from B to A, which is 2 units). The intersection of these arcs (above and below) will give two points, say, E and F. The line through E and F is the perpendicular to PA at A, which is the tangent line.

But the problem's step 2 says "Construct a perpendicular to line PA at point A". So using the compass method for perpendicular to a line at a point:

- Step 1: Given line PA and point A on it.
- Step 2: With A as center, draw an arc intersecting PA at two points, X and Y (on PA, X is towards P, Y is away from P, or vice versa, but distance from A to X = A to Y).
- Step 3: With X as center, draw an arc above PA (radius > AX).
- Step 4: With Y as center, draw an arc above PA (same radius as step 3), intersecting the arc from step 3 at point Z.
- Step 5: Similarly, draw an arc below PA from X and Y, intersecting at point W.
- Step 6: The line ZW is the perpendicular to PA at A, which is the tangent line.

So the construction steps are:

1. Ensure PA is a diameter (so the line through P and A is the diameter, center is midpoint of PA).
2. Construct the perpendicular to PA at A (using compass and straightedge), which will be the tangent line.

The final constructed tangent line is the line perpendicular to PA at A, as shown in the diagram (the slanted line through A, perpendicular to PA).

Answer:

The tangent to the circle at point A is constructed by drawing a line perpendicular to the diameter PA at point A (using compass and straightedge construction for perpendicular to a line at a point), which follows the property that a tangent to a circle is perpendicular to the radius (or diameter, in this case) at the point of contact.