Perpendicular to a given line through a given point | Maths

Starter quiz

This quadrilateral is a kite. Find the value of angle $a$ , in degrees.

'100' ✓
This diagram shows an accurate construction of a perpendicular bisector. YZ has a length of 8.4 cm. Which of these is the smallest value of $r$ that can be used in this construction?

2.1 cm

4.3 cm ✓

8.4 cm

16.8 cm

8.4 $\pi$ cm
Match the sentence to the diagram.

a

⇔

A bisector to line segment QV ✓

b

⇔

A perpendicular to line segment QV ✓

c

⇔

A perpendicular bisector to line segment QV ✓

d

⇔

An intersecting line to line segment QV ✓
This is an incomplete construction to find the perpendicular bisector of YZ. Which of these steps is missing from this construction?

A circle with centre at point Y.

A line segment connecting the intersection to point Z.

An arc below YZ with centre at point Z. ✓

An arc above YZ with centre at point Z.

A line through two pairs of intersecting arcs. ✓
This is an accurate construction to draw a rhombus. What is the perimeter (in cm) of the rhombus that can be drawn from this construction?

'56' ✓
Sofia starts her construction of a perpendicular bisector to EF. She is about to move the pair of compoasses to construct a second circle. Which of these statements are true?

Sofia needs to place the compass needle at the midpoint of EF.

Sofia should draw a circle with centre at the intersection of EF and the circle.

Sofia needs to draw an arc with centre at point F next.

Sofia should adjust the compass width so that both circles drawn are congruent. ✓

Currently, the next circle Sofia draws will be smaller than the first. ✓

Exit quiz

The line segment EF has been shortened by constructing a circle of radius 8 cm, with a centre at point K. What is the length (in cm) of the shortened line segment?

'16' ✓
Which of these diagrams shows the most appropriate first step when constructing the perpendicular to line segment TU through point Z?

a

b

c

d ✓
Izzy has shortened line segment LM by drawing a circle with centre at point Z. Izzy wants to draw four arcs of radius $k$ cm to construct a perpendicular. What value must $k$ be greater than?

4 cm

6 cm

12 cm ✓

14 cm

28 cm
A line segment is extended so that a perpendicular can be constructed through point Z. Which of these diagrams shows a suitable extension of the line segment?

a

b ✓

c

d
Which of these four circles can help in the construction of the perpendicular to line segment WU through point Z?

a

b ✓

c ✓

d
Match the values to their statements.

14 cm

⇔

the shortest distance from X to an extension of JK ✓

15 cm

⇔

the shortest distance from X to line segment JK ✓

17 cm

⇔

the radius of circle with centre at point X ✓

28 cm

⇔

the shortest distance from X to point J ✓

Worksheet

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Lesson Details

Key learning points

A rhombus can be constructed from two congruent isosceles triangles.
This can be used to construct a perpendicular to a line through a point.
This perpendicular has special properties that can be investigated.
Shortest distances can be found using constructions with perpendicular lines.

Common misconception

You can't accurately draw the perpendicular to a point on a line segment close to its endpoint.

The perpendicular to a line segment through any point can be found; some line segments just require extending in length first.

Keywords

Bisect - To bisect means to cut or divide an object into two equal parts.
Rhombus - A rhombus is a parallelogram where all sides are the same length.