Construction of Perpendicular Lines: Formula, Method, Sample Questions

Ans. A kite is a symmetrical shape. As a result, it has two equal and opposite angles. Its diagonals aren't all the same length, but the larger one divides the shorter one in half at right angles. There are four right angles at the intersection point because the two diagonals of the kite AD and BC connect at right angles or 90 degrees.

Ans. Construction steps are as follows:

1. Draw an AB line.
2. Join PQ with a point Q on AB and a point P outside AB.
3. Draw on the arc to cut AB at X and PQ at Z with Q as the centre and any radius.
4. Draw an arc cutting QP at Y with P as the centre and the same radius.
5. Draw an arc to cut the preceding arc at E, with Y as the centre and the radius equal to XZ.
6. To get the desired line, join PE and produce it on both sides.

Ans. A perpendicular bisector is a line segment that meets another line segment perpendicularly and divides it into two equal-sized halves. A rule, a compass, and a pencil can be used to draw a perpendicular bisector. When two lines intersect at 90 degrees or at right angles, they are perpendicular to each other. On the other hand, a bisector is a line that splits a line into two halves. A perpendicular bisector of a line segment XY intersects it at 90 degrees and divides it into two halves.

Ans. Properties of a perpendicular bisector are:

1. Creates two congruent segments from a line segment or a line.
2. Divides a triangle's sides into congruent sections.
3. They form a 90-degree angle with the line being bisected.
4. They cross the line segment exactly in the middle.
5. The circumcenter of a triangle is the place where the perpendicular bisectors intersect.
6. They meet inside an acute triangle, outside the triangle in an obtuse triangle, and at the hypotenuse in right triangles.
7. The perpendicular bisector's points are equidistant from both ends of the bisected segment.
8. For a given line segment, there can only be one.

Ans. A ruler and a compass can be used to make a perpendicular bisector on a line segment. The built perpendicular bisector separates a line segment into two equal pieces halfway, resulting in two congruent line segments.

Ans. A perpendicular bisector of a triangle is a line segment that bisects its sides while remaining perpendicular to them. They do not have to travel through the triangle's vertices, but they must pass through the midway of the sides. The perpendicular bisector of the triangle's sides is perpendicular to the middle of the triangle's sides.

Ans. When two lines connect at a 90° angle, they are perpendicular. Because the diagonals of a rhombus intersect at a 90° angle, we can say that the diagonals of a rhombus are perpendicular to one another. Because the diagonals of a rectangle do not intersect at a 90° angle, we can say that the diagonals of a rectangle are not perpendicular to one another.

Ans. To create the perpendicular bisector of a line segment AB = 10 cm. Construction steps are as follows:

1. Construct two arcs on both sides of the line segment AB and label them P and M, using A and B as centres and a radius greater than (1/2)AB.
2. Join P and M. 
3. Allow line PM to cut the AB portion of the line at Q.
4. The requisite perpendicular bisector AB is then line PMQ.

Ans. Lines that do not intersect and are always the same distance apart are called parallel lines. The symbol || is used to represent two parallel lines. Here are a few examples:

• The straight ladder's steps
• A rectangle's two opposite sides

Perpendicular lines are those that make a straight angle when they intersect. The symbol ⊥ is used to represent two perpendicular lines. Here are a few examples:

• The corner of two walls
• The letter "L"

Ans. 1. Four lines make up the square. A square has four right angles since each line is perpendicular to two other lines. The line AB is parallel to the line BC. BC, like CD, AB, and so on, is perpendicular to each other. To clarify, a square has right angles, and a square formed by drawing two more lines in the corner of a right angle is also used to represent a right angle.

2. A trapezium is a quadrangle in which one pair of opposite sides is parallel to one other while the other pair is not. The bottom of a trapezium is normally drawn using one of the parallel edges. The bases are the parallel edges, and the sides are the non-parallel edges. So, there are no perpendicular lines. Special trapeziums come in a variety of shapes and sizes. One side of a "right trapezium" is perpendicular to the bases, resulting in two right angles. So, the lines are perpendicular.

3. Right angles can also be found in rectangles. A rectangle, like a square, has four right angles. This means it has lines that are perpendicular to each other.

4. Rhombus has equal lines but no perpendicular lines as clearly shown in the figure. The rhombus has diagonals which bisect each other at right angles, so the diagonals are perpendicular to each other, whereas the sides are not.

5. A right triangle has perpendicular lines as shown in the figure.

6. An equilateral triangle has all angles equal to 600, so it does not have perpendicular lines.