Lesson 1: Geometrical Constructions | State Board · Class 7 · Mathematics

Bisectors, incentre and circumcentre

From daily life

To cut a pizza into two equal pieces we cut straight through the middle. That cutting line is a "bisector".

Angle bisector: a ray that splits an angle into two equal parts. In the figure ∠AOB = 72°; the bisector OP splits it into 36° and 36°.

Bisecting a 90° angle gives 45° each.
Bisecting a 110° angle gives 55° each.

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Bisectors, incentre and circumcentre

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Perpendicular bisector: a line that crosses a segment at its midpoint at 90°. It cuts the segment into two equal halves.

If PS = 8 cm then PM = MS = 4 cm and ∠CMP = 90°.
For a 10 cm segment each half is 5 cm.

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Bisectors, incentre and circumcentre

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The bisectors of all three angles of a triangle meet at one point: it is called the incentre (I).

I is always inside the triangle.
The perpendiculars IA, IB, IC from I to the sides are equal, so a circle drawn from I touches all three sides.

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Bisectors, incentre and circumcentre

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The perpendicular bisectors of the three sides pass through one point: it is called the circumcentre (C).

C is at the same distance from all three vertices: CX = CY = CZ. So a circle drawn from C passes through all three vertices.
Example: a water tank that must be equally far from three villages should be built at the circumcentre of the triangle formed by the villages.

Where is the circumcentre?

Type of triangle	Circumcentre
Acute-angled (all angles below 90°)	inside the triangle
Right-angled	at the midpoint of the hypotenuse
Obtuse-angled (one angle above 90°)	outside the triangle

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Bisectors, incentre and circumcentre

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Remember

the incentre (I) is always inside. The circumcentre (C) is inside, on the hypotenuse or outside, depending on the triangle.

Easy trick

I = Inside, "incentre" goes with angle bisectors. C = Circle through the Corners, "circumcentre" goes with perpendicular bisectors.

Common mistake

treating the two points as the same. Angle bisectors meet at I; perpendicular bisectors of the sides meet at C.

Spot the difference

Point	Incentre (I)	Circumcentre (C)
What meets?	three angle bisectors	three perpendicular bisectors of the sides
Equal distance from	the three sides	the three vertices
Position	always inside	inside / on hypotenuse / outside

Practice questions

Easy

1Bisecting a 90° angle, how big is each part?
2A perpendicular bisector cuts a 10 cm segment into what lengths?
3What is the meeting point of the three angle bisectors called?

Medium

4Which point of a triangle is at equal distance from its three sides?
5A tank must be equally far from three villages. Which point will you choose?
6Where is the circumcentre of a right-angled triangle?

Challenging

7The distance from the circumcentre to vertex X is 5 cm. What is the diameter of the circle through the vertices?
8Where are I and C in an obtuse-angled triangle?

Show answers (try them yourself first)

145° 2) 5 cm and 5 cm 3) incentre
4incentre (I) 5) circumcentre (C) 6) at the midpoint of the hypotenuse
7CX = CY = CZ = 5 cm is the radius, so the diameter is 10 cm.
8I is inside the triangle; C is outside it.

Constructing a triangle from three sides

From daily life

Before building a house, a plan is drawn on paper. In the same way, before drawing a triangle we draw a rough figure; it helps to plan the steps.

Rule for the rough figure: show the given measures and draw the longest side longest.

When is it not enough? With only two sides given, many triangles are possible; a third piece of information is needed.

Rule: given three sides, take the longest as the base; draw arcs for the other two sides; they cross at the third vertex.

Steps (AB = 9 cm, BC = 6 cm, CA = 7 cm)

1) Draw the base AB = 9 cm.
2) Open the compass to 7 cm, put the point at A and draw an arc.
3) Open it to 6 cm, put the point at B and cut the first arc. The crossing point is C.
4) Join AC and BC. ΔABC is ready.

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Constructing a triangle from three sides

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Solved examples

Easy: DE = 7 cm, EF = 5 cm, DF = 4.5 cm. Base DE = 7 cm; arc of 4.5 cm from D; arc of 5 cm from E; the crossing point is F.
Medium: equilateral triangle, side 5.5 cm. Base 5.5 cm; arcs of 5.5 cm from both ends; the crossing point is the third vertex.
Exam type: isosceles triangle, base 4 cm, equal sides 5 cm. Base 4 cm; arcs of 5 cm from both ends.

Condition on the sides: the sum of any two sides must be greater than the third. For example 3 cm, 2 cm and 7 cm cannot form a triangle because 3 + 2 = 5 is less than 7; the arcs never meet.

Remember

taking the longest side as the base makes the construction easy. If the arcs are drawn on the opposite side of the base, you get a mirror-image triangle of the same size.

Common mistake

changing the compass opening or putting the point at the wrong place. One arc = one opening and one centre.

Spot the difference

Type of triangle	Sides	Construction
Equilateral	all three equal	both arcs have radius = side
Isosceles	two equal	both arcs have the same radius
Scalene	all different	the two arcs have different radii

Practice questions

Easy

1Given three sides, which side do you take as the base?
2How many arcs are drawn in the construction?
3Can a triangle with sides 7 cm, 9 cm and 12 cm be drawn?

Medium

4Can a triangle with sides 3 cm, 4 cm and 8 cm be drawn?
5What radius is used for the arcs of an equilateral triangle of side 5.5 cm?
6Can an isosceles triangle with base 11 cm and equal sides 6 cm be drawn?

Challenging

7Why can a triangle with sides 2.5 cm, 3 cm and 6 cm not be drawn?
8What is the perimeter of a triangle with sides 8 cm, 6.5 cm and 5.5 cm?

Show answers (try them yourself first)

1the longest side 2) two 3) Yes; 7 + 9 = 16 > 12
4No; 3 + 4 = 7 is less than 8. 5) 5.5 cm 6) Yes; 6 + 6 = 12 > 11
72.5 + 3 = 5.5 is less than 6; the arcs do not meet.
88 + 6.5 + 5.5 = 20 cm

Constructing a triangle from sides and angles

From daily life

Once the two blades of a pair of scissors and the angle between them are fixed, the shape of the scissors is fixed. In the same way, two sides and the angle between them fix a triangle.

Type (a): two sides and the angle between them

Steps (PQ = 6.5 cm, ∠P = 68°, PR = 5 cm)

1) Draw the base PQ = 6.5 cm.
2) At P make an angle of 68° and draw ray PG.
3) Open the compass to 5 cm, put the point at P and cut PG at R.
4) Join QR. You may draw the ray on the opposite side of the base as well.

Type (b): two angles and the side between them

Steps (PQ = 7 cm, ∠P = 55°, ∠Q = 65°)

1) Draw PQ = 7 cm.
2) Make 55° at P and 65° at Q on the same side.
3) The two rays meet at R. ΔPQR is ready; ∠R = 180° − 55° − 65° = 60°.

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Constructing a triangle from sides and angles

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Solved examples

Easy: two angles 50° and 60° give the third angle = 180° − 110° = 70°.
Medium: ∠D = 56°, ∠E = 74°, side DF = 6.5 cm. First ∠F = 180° − 130° = 50°. Now DF lies between ∠D and ∠F, so the "two angles and the side between them" method works.
Exam type: ∠X = 35°, ∠Y = 75°. The third angle ∠Z = 180° − 110° = 70°. Any given side can then be used as the side between two angles.

Remember

the angle must be the one between the two sides; otherwise the construction is different (sometimes two triangles). If the given side is not between the two angles, first find the third angle.

Shortcut

if two angles are known, the third is 180° − (sum of the two).

Common mistake

making the angle at the wrong end. The 68° angle lies between the base PQ and ray PG, at P, not at Q.

Spot the difference

Point	Two sides + angle between	Two angles + side between
What is drawn first?	the base	the base
Compass needed?	yes, for one arc	no; only a protractor
Third vertex	cut the ray with an arc	where the two rays meet

Practice questions

Easy

1Two angles are 50° and 60°. What is the third?
2In the "two sides and angle" construction, must the angle be between the two sides?
3With two angles and the side between them, how do we get the third vertex?

Medium

4∠A = 45°, ∠B = 85°. Find ∠C.
5Why can a triangle with angles 100° and 80° not be drawn?
6∠P = 38°, ∠R = 72° and side PQ is given. Find ∠Q. Between which angles is PQ?

Challenging

7∠D = 56°, ∠E = 74° and DF = 6.5 cm. How will you construct it?
8Two angles are 55° and 65° with a 7 cm side between them. Find the third angle.

Show answers (try them yourself first)

170° 2) yes 3) where the two rays meet
4180° − 130° = 50° 5) 100° + 80° = 180°, so the third angle would be 0°. 6) ∠Q = 70°; PQ lies between ∠P and ∠Q.
7∠F = 50°; DF lies between ∠D and ∠F, so take it as the base and make ∠D = 56° and ∠F = 50°.
8180° − 120° = 60°

Right-angled triangles and special cases

From daily life

A ladder leaning on a wall makes a right angle with the ground. The ladder is the longest side, the hypotenuse.

Hypotenuse: the side that faces the right angle. It is the longest side of the triangle.

Construction (hypotenuse 7 cm and one side MN = 5 cm, ∠M = 90°)

1) Draw the base MN = 5 cm.
2) At M draw a ray MT at 90°.
3) Open the compass to 7 cm, put the point at N and cut MT at L.
4) Join LN. LN is the hypotenuse.

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Right-angled triangles and special cases

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Solved examples

Easy: ∠B = 90°, hypotenuse AC = 6.5 cm, BC = 3.9 cm. Draw BC, a 90° ray at B, and an arc of 6.5 cm from C.
Medium: a right-angled triangle with hypotenuse 6 cm and a side of 8 cm cannot exist, because the hypotenuse is the longest side.
Exam type: one angle of a right-angled triangle is 90°, so the other two angles together make 90°.

Special case 1: no triangle can be drawn.

∠A = 95° and ∠B = 120°: the sum is 215°, more than 180°.
Sides 2.5 cm, 3 cm and 6 cm: the sum of two sides is less than the third.

Special case 2: two different triangles. With BC = 9.5 cm, CA = 6.5 cm and ∠B = 38°, the arc from C cuts the ray at two points (A and A′), giving two triangles.

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Right-angled triangles and special cases

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Special case 3: only three angles given. If the angles are 60°, 50°, 70° and no side is given, many triangles fit. They have the same shape but different sizes.

Remember

the hypotenuse is the longest side. The angles of a triangle always add up to 180°.

Common mistake

putting the compass point at the wrong place. The arc for the hypotenuse is drawn from the end that is not the right angle (N), not from M.

Spot the difference

Given information	Result
two angles add up to 180° or more	no triangle
two sides and an angle that is not between them	sometimes two triangles
only three angles	many triangles (same shape)
hypotenuse and one side (right-angled)	one triangle

Practice questions

Easy

1Which is the longest side of a right-angled triangle?
2How many degrees is the right angle?
3Where is the compass point placed when drawing the arc for the hypotenuse?

Medium

4Can a right-angled triangle with hypotenuse 6 cm and a side of 8 cm exist?
5Can a triangle have ∠A = 90° and ∠B = 95°?
6If the arc cuts the ray at two points, how many triangles do we get?

Challenging

7How many triangles fit the three angles 60°, 50°, 70° alone?
8A right-angled triangle has one angle of 35°. Find the third angle.

Show answers (try them yourself first)

1hypotenuse 2) 90° 3) at the end that is not the right angle (e.g. N)
4No; the hypotenuse is the longest side. 5) No; 90° + 95° = 185° > 180°. 6) two
7many; same shape but different sizes. 8) 180° − 90° − 35° = 55°

Congruence of segments, angles and circles

From daily life

Two copies of the same note placed one over the other fit exactly. We say they are congruent. The symbol is ≅.

Congruent segments: segments of the same length are congruent.

Easy: seg EF = 5 cm and seg GH = 5 cm, so seg EF ≅ seg GH.
Rule: if seg EF ≅ seg GH then seg GH ≅ seg EF. And if seg GH ≅ seg JK then seg EF ≅ seg JK.

Congruent angles: angles of the same measure are congruent. It does not matter whether the arms are short or long.

In the figure ∠ABC ≅ ∠PQR (40° each). ∠STU = 55°, so it is not congruent to them.

Congruent circles: circles of the same radius are congruent.

Circle P ≅ circle Q (both 1.8 cm), but circle R (2.2 cm) is not congruent to them.
If the diameter is given, halve it to get the radius: diameter 8 cm means radius 4 cm.

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Congruence of segments, angles and circles

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Remember

for segments compare lengths, for angles compare measures, for circles compare radii. The length of the arms does not affect congruence of angles.

Common mistake

thinking an angle with longer arms is bigger. A 40° angle stays 40° however long the arms are drawn.

Spot the difference

Object	Congruent when	Example
Segment	same length	5 cm and 5 cm
Angle	same measure	40° and 40°
Circle	same radius	1.8 cm and 1.8 cm

Practice questions

Easy

1Are segments of 5 cm and 5 cm congruent?
2Is a 40° angle congruent to another 40° angle with different arm lengths?
3Are circles of radius 3 cm and 3 cm congruent?

Medium

4If seg AB ≅ seg CD and seg CD ≅ seg EF, what can you say about AB and EF?
5Are a circle of diameter 8 cm and a circle of radius 4 cm congruent?
6Are circles of radius 2 cm and 2.5 cm congruent?

Challenging

7One 40° angle has 3 cm arms and another 40° angle has 6 cm arms. Are they congruent?
8Are segments of 7.2 cm and 7.3 cm congruent?

Show answers (try them yourself first)

1yes 2) yes; same measure 3) yes
4AB ≅ EF 5) yes; diameter 8 means radius 4 6) no; the radii differ
7yes; the measures are equal, so arm length does not matter. 8) no; the lengths differ.