Construct an isosceles triangle PQR in which PQ = PR, base QR = 5.6 centimetres, angle PQR = 75 degrees using a ruler and a pair of compasses. Draw its lines/line of symmetry.
Since the angles opposite to the equal sides of an isosceles triangle are equal, we draw a 75 degree angle at R. The point at which the rays from Q and R meet is named P. We will find that PQ = PR.
To construct an isosceles triangle PQR with given conditions, follow these steps:
1. Start by drawing a straight line segment QR using a ruler. According to the given information, the length of QR should be 5.6 centimeters.
2. Use a compass to mark the starting point of the triangle on the line segment QR. This point will be the vertex, Q.
3. Open the compass wider than half of QR. Place the compass on point Q and draw an arc that intersects QR on both sides.
4. Without adjusting the compass width, place the compass on the intersection points of the arc and QR. Draw two arcs that intersect each other inside the triangle.
5. With the compass still set to the same width, place the compass at one of the intersection points and draw an arc that cuts the previous arc (drawn in step 4) at another point. Label this point as P.
6. Place the compass on point P and draw an arc that intersects QR at the other intersection point. Label this point as R.
7. Complete the isosceles triangle by drawing line segments PQ and PR.
To draw the lines of symmetry:
1. Bisect the angle PQR: Use a compass to draw arcs from points P and R, cutting the angle PQR. Label the points of intersection with the angle as X and Y.
2. With a ruler, draw a line segment from point Q through the point of intersection X. This line segment, QR, will be the line of symmetry for the triangle.
Now, you have successfully constructed the isosceles triangle PQR and its line of symmetry QR.