1
A pure construction in this chapter uses which two tools?
💡 Hint:
A graduated scale is not required.
Explanation:
A straight edge and a compass.
Class 9 · Maths · Chapter 11 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Constructions
Constructions
How to use this page:
1. Read — Construction 11.1 · Construction 11.2 · Construction 11.3 · Exercise 11.1 · Construction 11.4 · Construction 11.5 · Construction 11.6 · Exercise 11.2, diagram, worked example, board tip
2. Check — each lesson has its own questions; the number follows the lesson
3. Mastery ★ — all of that lesson correct. Redo the wrong ones
4. The memory figure shows an equilateral triangle made by two equal-radius arcs at the end of a ray, so the angle drawn is 60°.
This page follows the Bihar textbook. Ganita Manjari uses a different order. Progress stays in this browser.
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- 1 What a construction is, and the geometry box
- 2 Construction 11.1 — the bisector of an angle
- 3 Construction 11.2 — the perpendicular bisector
- 4 60° and the angles of Exercise 11.1
- 5 Construction 11.4 — base, angle, and the sum of the sides
- 6 Construction 11.5 — the difference of the sides, two cases
- 7 Construction 11.6 and Exercise 11.2
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
What a construction is, and the geometry box
रचना का अर्थ और ज्यामिति पेटी · Textbook 11.1 · straight edge · compass
An accurate figure uses two toolsNotes
Figures in earlier chapters were often only for reasoning. A map, a tool or a road plan must be accurate. The box holds a scale, two set squares, dividers, a compass and a protractor.
A geometrical construction in this chapter uses only an ungraduated straight edge and a compass. Where a measurement is also asked, a scale and a protractor may be used. The protractor is a check. It does not replace a compass step.
Worked exampleExample
Question: Find one angle of an equilateral triangle in degrees. This is the reason for the 60° construction.
Formula: each angle of an equilateral triangle = 180° ÷ 3.
Substitution: 180° ÷ 3 = 60°.
Unit: degree.
10-second revision
- Construction = straight edge + compass
- A protractor is for checking
- An equilateral angle is 60°
Board tip · BSEBBoard tip
In the answer write which step uses the compass and which line is only the straight edge.
Board tip · CBSEBoard tip
Do not call a 60° drawn with a set square the construction. The reason is the equilateral triangle.
Check your understandingall correct = mastery ★
2
Every construction step is completed by measuring the angle with a protractor.
Explanation:
False — the protractor is for checking. The steps use a compass and a straight edge.
3
Each angle of an equilateral triangle is ______ degrees.
💡 Hint:
Divide 180 by 3.
Explanation:
60°. The unit is the degree.
4
The 30°-60°-90° set square in the geometry box —
💡 Hint:
The box and a construction are not the same thing.
Explanation:
It is in the box. A pure construction asks for a straight edge and a compass.
5
Write two differences between a rough figure and a construction. Also give one reason for 60°.
Model answer (board format)
A rough figure only shows the situation. A construction is built to the given measures with a straight edge and a compass. The reason for 60° is an equilateral triangle: 180° ÷ 3 = 60°.
Check your answer:
Construction 11.1 — the bisector of an angle
रचना 11.1 — कोण समद्विभाजक · Textbook 11.2 · Construction 11.1 · SSS
One arc, then two arcs larger than half the gapNotes
Angle ABC is given. With centre B and any radius, draw an arc cutting BA and BC at E and D. Then with centres D and E and equal radii greater than half of DE, let the arcs meet at F and draw BF. That is the bisector.
Reason: BE = BD, EF = DF, and BF is common. SSS makes the triangles congruent, so angles EBF and DBF are equal. CPCT says this.
Do Construction 11.1 yourselfActivity
Draw any angle. One arc from the vertex, then arcs greater than half the gap from both cuts. Draw the ray through the meeting point. Check the two halves with a protractor. The check does not replace the steps.
Worked exampleExample
Question: How large is each part made by the bisector of an 80° angle?
Formula: each half = given angle ÷ 2.
Substitution: 80° ÷ 2 = 40°.
Unit: degree.
10-second revision
- The second radius is greater than half of DE
- SSS and then CPCT
- Half of 80° is 40°
Board tip · BSEBBoard tip
The first radius is “any”, but forgetting to write that the second arcs are greater than half loses marks.
Board tip · CBSEBoard tip
Do not write only “BF is the bisector”. Name the three equal sides of SSS.
Check your understandingall correct = mastery ★
1
The radius of the second arcs of an angle bisector must be —
💡 Hint:
The arcs must meet.
Explanation:
Greater than half of DE.
2
A bisector splits the angle into two equal parts.
Explanation:
True — CPCT makes the two angles equal.
3
Bisecting 70° makes each part ______ degrees.
💡 Hint:
70 ÷ 2.
Explanation:
35°.
4
Write the three steps of Construction 11.1 and one line of SSS.
Model answer (board format)
An arc from the vertex, then two arcs greater than half the gap, then the ray through the meeting point. BE = BD, EF = DF, BF common, so SSS.
Check your answer:
Construction 11.2 — the perpendicular bisector
रचना 11.2 — लंब समद्विभाजक · Textbook 11.2 · Construction 11.2 · 90°
From both ends, on both sides of the lineNotes
The perpendicular bisector of segment AB is required. With centres A and B and radius greater than half of AB, draw arcs on both sides. Join the cuts P and Q. PQ cuts AB at M. PMQ is the required perpendicular bisector.
First SSS makes the large triangles congruent. Then SAS gives AM = BM and equal angles. They are a linear pair, so each is 90°.
Do Construction 11.2 yourselfActivity
Take a segment of 6 cm. From both ends draw arcs greater than half the length, on both sides. Join the meeting points. Measure the cut: both pieces should be 3 cm and the angle 90°.
Worked exampleExample
Question: AB = 8 cm. The perpendicular bisector meets it at M. Find AM and angle PMA.
Formulas: AM = AB ÷ 2. Each of two equal angles in a linear pair = 180° ÷ 2.
Substitution: AM = 8 ÷ 2 = 4 cm. The angle = 180° ÷ 2 = 90°.
Units: centimetre and degree.
10-second revision
- Radius greater than half of AB
- AM = BM and the angle is 90°
- 90° comes from the linear pair
Board tip · BSEBBoard tip
Arcs on only one side are an incomplete construction. The book says both sides.
Board tip · CBSEBoard tip
When you write 90°, show 180° ÷ 2. Only “it is perpendicular” is half the reason.
Check your understandingall correct = mastery ★
1
The radius of the arcs for a perpendicular bisector is —
💡 Hint:
The arcs must meet on both sides.
Explanation:
Greater than half of AB.
2
A perpendicular bisector cuts the segment in half at 90°.
Explanation:
True — AM = BM and each angle is 90°.
3
If AB = 10 cm, the middle piece AM is ______ cm.
💡 Hint:
10 ÷ 2.
Explanation:
5 cm.
4
The first congruence rule used here is —
💡 Hint:
Three sides.
Explanation:
SSS. SAS comes after that.
5
PQ cuts the segment at one endpoint, not in the middle.
Explanation:
False — the cut M is the midpoint. AM = BM.
6
How do AM and 90° come out on an 8 cm segment? Write two formulas.
Model answer (board format)
AM = AB ÷ 2 = 8 ÷ 2 = 4 cm. The two equal angles are a linear pair, so each is 180° ÷ 2 = 90°.
Check your answer:
60° and the angles of Exercise 11.1
60° और अभ्यास 11.1 के कोण · Construction 11.3 · Exercise 11.1
One radius, used twiceNotes
From the end A of ray AB draw an arc cutting the ray at D. From D, with the same radius, cut the earlier arc at E. Draw AE. Triangle EAD is equilateral, so the angle is 60°.
Further angles come from this 60°. Half is 30°, and half of that is 15°. 22.5° is half of half a right angle. 90° can be built by adding 60° and 30°, or by halving 180°. 45° is half a right angle.
Exercise 11.1 — the angles in this orderActivity
Do not change the book order. First justify 90°, then justify 45°. Then 30°, 22.5° and 15°. Then 75°, 105° and 135° with a protractor check. Last, an equilateral triangle on a given side, with a reason. 75° = 60° + 15°. 105° = 60° + 45°. 135° = 90° + 45°.
Worked exampleExample
Question: How will you obtain 15° from 60°?
Formula: 15° = 60° ÷ 4. Two bisections.
Substitution: 60° ÷ 2 = 30°, then 30° ÷ 2 = 15°.
Unit: degree.
10-second revision
- 60° = an equilateral angle
- 15° = a quarter of 60°
- 75°, 105° and 135° by adding
Board tip · BSEBBoard tip
Do not write 22.5° as 22°. It is half of 45°. The third part of Exercise 11.1 asks for this.
Board tip · CBSEBoard tip
Do not draw 75° with a protractor and leave the reason blank. Show 60° + 15°.
Check your understandingall correct = mastery ★
1
The 60° of Construction 11.3 appears because the triangle is —
💡 Hint:
All three sides are one radius.
Explanation:
It is equilateral, so 60°.
2
45° is half of a right angle.
Explanation:
True — 90° ÷ 2 = 45°.
3
Halving 60° twice leaves ______ degrees.
💡 Hint:
30, then half.
Explanation:
15°.
4
One sum that builds 135° is —
💡 Hint:
The fourth part of Exercise 11.1.
Explanation:
90° + 45° = 135°.
5
How will you build 90° and 75° by combining pieces of 60°? Keep the unit as the degree.
Model answer (board format)
30° = 60° ÷ 2. 90° = 60° + 30°. 15° = 30° ÷ 2. 75° = 60° + 15°.
Check your answer:
Construction 11.4 — base, angle, and the sum of the sides
रचना 11.4 — आधार, कोण और भुजाओं का योग · Textbook 11.3 · Construction 11.4
Cut BD equal to the whole sumNotes
The base BC, angle B, and AB + AC are given. Draw the base, make the given angle at B, and cut BD = AB + AC on the ray. Join DC and make angle DCY equal to angle BDC. CY cuts the ray at A.
The other way: the perpendicular bisector of CD cuts BD at A. Then AD = AC. The construction is possible only when the sum is greater than the base. If AB + AC ≤ BC, the triangle does not exist.
The caution in Construction 11.4Activity
Compare the sum and the base first. Cut BD only when the sum is larger. The second angle, equal to the base angle of the small triangle, brings A onto the ray, because then AC = AD. If two sides and a non-included angle are given, the triangle is not always unique. That is the book’s caution.
Worked exampleExample
Question: The base is 6 cm and AB + AC = 10 cm. Is Construction 11.4 possible?
Formula: the construction is possible when AB + AC > BC.
Substitution: 10 cm > 6 cm, so it is possible. BD must be cut as 10 cm.
Unit: centimetre.
10-second revision
- BD = AB + AC
- The perpendicular bisector gives AD = AC
- Impossible when the sum ≤ the base
Board tip · BSEBBoard tip
Do not cut BD equal to only one side. The book cuts the whole sum.
Board tip · CBSEBoard tip
The impossible case comes as its own sentence in an exam. Write ≤, not only “smaller”.
Check your understandingall correct = mastery ★
1
If AB + AC equals the base, Construction 11.4 is —
💡 Hint:
The sum must exceed the base.
Explanation:
Not possible. Not even when they are equal.
2
If A lies on the perpendicular bisector of CD, then AD = AC.
Explanation:
True — this is the second method.
3
If the base is 7 cm and the sum is 13 cm, BD is cut as ______ cm.
💡 Hint:
The whole sum.
Explanation:
13 cm.
4
The base is 9 cm and AB + AC = 9 cm. Why does the construction stop? Write the rule and the substitution.
Model answer (board format)
The rule is AB + AC > BC. Here 9 cm = 9 cm, so the sum is not greater than the base. The construction is not possible. The unit is the centimetre.
Check your answer:
Construction 11.5 — the difference of the sides, two cases
रचना 11.5 — भुजाओं का अंतर, दो केस · Textbook 11.3 · Construction 11.5
The longer side chooses the caseNotes
The base BC, angle B, and the difference of the other sides are given. Case one: AB > AC, so AB − AC is given. Cut BD equal to this difference on ray BX. The perpendicular bisector of DC cuts the ray at A. Then AD = AC, so BD = AB − AC.
Case two: AC > AB. Cut the difference on the line extended opposite the base. The perpendicular bisector again gives A. Do not swap the steps of one case into the other.
The two cases in separate figuresActivity
In one figure the difference lies on the ray. In the other it lies on the side opposite the base. In both, PQ is the perpendicular bisector of DC. Question 2 of Exercise 11.2 is the first case and question 3 is the second case. Keep them in that order for now. The full exercise is worked in the next lesson.
Worked exampleExample
Question: AB − AC = 3.5 cm and AB is the longer side. How long is BD, and on which side?
Formula: in case one, BD = AB − AC, on ray BX.
Substitution: BD = 3.5 cm, on the ray of the angle.
Unit: centimetre.
10-second revision
- If AB > AC, the difference is on the ray
- If AC > AB, use the opposite direction
- In both, AD = AC
Board tip · BSEBBoard tip
The side written first in the subtraction is the longer one. AB − AC means AB is longer.
Board tip · CBSEBoard tip
Do not push both cases into one figure. An examiner looks for the case separately.
Check your understandingall correct = mastery ★
1
The cases of Construction 11.5 are —
💡 Hint:
The difference can be in either order.
Explanation:
Two cases.
2
If AB > AC, BD is cut on ray BX.
Explanation:
True — this is case one.
3
If the difference is 2 cm and the first side is longer, BD is ______ cm.
💡 Hint:
Equal to the difference.
Explanation:
2 cm.
4
If AC > AB, the difference is cut —
💡 Hint:
Case two.
Explanation:
In the opposite direction.
5
Cutting in the same direction is enough for both cases.
Explanation:
False — the other longer side asks for the opposite direction.
6
Base QR, angle Q, and PR − PQ = 2 cm. Which case is it? Which piece, in place of BD, is 2 cm?
Model answer (board format)
Read Q as B and R as C. PR − PQ = AC − AB, so AC is longer. This is case two. The 2 cm piece is cut on the side opposite the base. The unit is the centimetre.
Check your answer:
Construction 11.6 and Exercise 11.2
रचना 11.6 और अभ्यास 11.2 · Construction 11.6 · Exercise 11.2
The first line equals the whole perimeterNotes
Two base angles and the perimeter AB + BC + CA are given. Draw XY equal to the perimeter. Make angle B at X and angle C at Y. The bisectors of the two angles meet at A. The perpendicular bisectors of AX and AY cut XY at B and C.
Reason: B lies on the perpendicular bisector of AX, so XB = AB. Likewise CY = AC. The sum leaves XY equal to the whole perimeter. The angles, doubled from the bisectors, become the given angles.
Exercise 11.2, all five questions in orderActivity
1) Base 7 cm, angle 75°, sum of the other two sides 13 cm. This is Construction 11.4. 2) Base 8 cm, angle 45°, AB − AC = 3.5 cm. Case one. 3) Base 6 cm, angle 60°, PR − PQ = 2 cm. Case two. 4) Angles 30° and 90°, perimeter 11 cm. Construction 11.6. 5) A right triangle, base 12 cm, sum of the hypotenuse and the other side 18 cm. This is the sum construction again.
Worked exampleExample
Question: The perimeter is 12 cm. How long is the first line of Construction 11.6?
Formula: XY = AB + BC + CA.
Substitution: XY = 12 cm.
Unit: centimetre.
10-second revision
- XY = the perimeter
- The two bisectors meet at A
- XB = AB and CY = AC
Board tip · BSEBBoard tip
In Exercise 11.2 name the case. Question 2 is on the ray, question 3 is in the opposite direction.
Board tip · CBSEBoard tip
Do not treat the perimeter line as one side. It is the sum of all three sides.
Check your understandingall correct = mastery ★
1
In Construction 11.6, XY equals —
💡 Hint:
The sum of the three sides.
Explanation:
The perimeter.
2
The perpendicular bisector of AX gives XB = AB.
Explanation:
True — B lies on that perpendicular bisector.
3
If the perimeter is 11 cm, XY is ______ cm.
💡 Hint:
The whole sum.
Explanation:
11 cm.
4
A right triangle with the base and the sum of the hypotenuse and the other side uses —
💡 Hint:
Question 5 of Exercise 11.2.
Explanation:
The sum construction, 11.4.
5
Write the five jobs of Exercise 11.2 in order, one line each. Repeating the measures is enough.
Model answer (board format)
1) 7 cm, 75°, sum 13 cm. 2) 8 cm, 45°, difference 3.5 cm on the ray. 3) 6 cm, 60°, difference 2 cm on the opposite side. 4) 30°, 90°, perimeter 11 cm. 5) Right-angled, base 12 cm, sum of hypotenuse and the other side 18 cm.
Check your answer:
❓ Full question bank — with answers and explanations — 66 questions
Pick a type. The 35 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ130 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA120 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/13Pick one option. A wrong try brings a hint.
1
The tools of a pure construction are —
💡 Hint:
The box has more things. The steps are these two.
Explanation:
A straight edge and a compass.
2
The second radius of an angle bisector is —
💡 Hint:
So that the arcs meet.
Explanation:
Greater than half.
3
A perpendicular bisector makes —
💡 Hint:
A linear pair.
Explanation:
90° and the midpoint.
4
The reason for 60° is —
💡 Hint:
180 ÷ 3.
Explanation:
An equilateral triangle.
5
15° is obtained as —
💡 Hint:
Halve twice.
Explanation:
60° ÷ 4 = 15°.
6
One sum for 75° is —
💡 Hint:
Exercise 11.1.
Explanation:
60° + 15°.
7
One sum for 135° is —
💡 Hint:
A right angle and its half.
Explanation:
90° + 45°.
8
If AB + AC equals the base, Construction 11.4 is —
💡 Hint:
The sum must be greater.
Explanation:
Not possible.
9
If AB − AC is given and AB is longer, BD is cut —
💡 Hint:
Case one.
Explanation:
On the ray.
10
If the given difference is AC − AB, cut —
💡 Hint:
Case two.
Explanation:
In the opposite direction.
11
In Construction 11.6, XY is —
💡 Hint:
The sum of the three.
Explanation:
The perimeter.
12
Question 5 of Exercise 11.2 uses —
💡 Hint:
The sum of the hypotenuse and the other side.
Explanation:
The sum construction.
13
One sum for 105° is —
💡 Hint:
Exercise 11.1.
Explanation:
60° + 45° = 105°.
True or false
0/81
A protractor can replace every step of a construction.
Explanation:
False — it is a check.
2
An angle bisector makes two equal angles.
Explanation:
True — SSS and CPCT.
3
A perpendicular bisector is complete with arcs on only one side.
Explanation:
False — the book says both sides.
4
22.5° is half of half a right angle.
Explanation:
True — 90° ÷ 4 = 22.5°.
5
A triangle still exists when AB + AC is less than the base.
Explanation:
False — the construction is not possible.
6
Both difference cases cut in the same direction.
Explanation:
False — the second case is in the opposite direction.
7
In Construction 11.6, XB = AB.
Explanation:
True — because of the perpendicular bisector.
8
An equilateral triangle on a given side is the last job of Exercise 11.1.
Explanation:
True — the fifth question.
Fill in the blanks
0/81
An equilateral angle is ______ degrees.
💡 Hint:
180 ÷ 3.
Explanation:
60.
2
Half of 80° is ______ degrees.
💡 Hint:
80 ÷ 2.
Explanation:
40.
3
If AB = 8 cm, AM is ______ cm.
💡 Hint:
8 ÷ 2.
Explanation:
4.
4
Half of 90° is ______ degrees.
💡 Hint:
Half a right angle.
Explanation:
45.
5
If the sum is 13 cm, BD is ______ cm.
💡 Hint:
The whole sum.
Explanation:
13.
6
If the difference is 3.5 cm, the cut piece in that case is ______ cm.
💡 Hint:
Equal to the difference.
Explanation:
3.5.
7
If the perimeter is 12 cm, XY is ______ cm.
💡 Hint:
XY = the perimeter.
Explanation:
12.
8
60° + 45° = ______ degrees.
💡 Hint:
Exercise 11.1.
Explanation:
105.
Match
0/21
Match the construction with its job.
Column B: A. Perimeter and two base angles · B. Halving an angle · C. Base, angle, and the sum of the sides · D. Perpendicular bisector
1. 11.1
2. 11.2
3. 11.4
4. 11.6
Explanation:
11.1 halves an angle, 11.2 is the perpendicular, 11.4 is the sum, and 11.6 is the perimeter.
2
Match the angle with its sum.
Column B: A. 90° + 45° · B. 60° ÷ 4 · C. 60° + 15° · D. 60° + 45°
1. 15°
2. 75°
3. 105°
4. 135°
Explanation:
15 is the quarter, 75 and 105 are sums, and 135 is a right angle plus a half.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): The 60° construction makes an equilateral triangle.
Reason (R): All three sides equal one and the same radius.
Explanation:
Both are true and R is the reason. 180° ÷ 3 = 60°.
2
Assertion (A): A perpendicular bisector makes 90°.
Reason (R): An equilateral triangle also has an angle of 60°.
Explanation:
Both are true, but R does not explain the 90°.
3
Assertion (A): Construction 11.4 fails when AB + AC ≤ the base.
Reason (R): In that situation it is enough to cut BD in half.
Explanation:
A is true. R is false — the construction stops. Halving BD does not build it.
4
Assertion (A): Both difference cases cut in the same direction.
Reason (R): If AC > AB, the difference is cut in the opposite direction.
Explanation:
A is false. R is true, case two.
5
Assertion (A): In Construction 11.6, XY equals the perimeter.
Reason (R): XB = AB and CY = AC, so the sum is the whole line.
Explanation:
Both are true and R explains A.
Coefficient practice
0/4These are not results of this maths chapter. They are only coefficient practice for H2 + O2 → H2O.
1
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). This is not a construction result.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). Match the hydrogen count.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
3
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). Match the oxygen count.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
4
This is coefficient practice, not a result of this maths chapter. Fill coefficients for H2 + O2 making H2O (blank = 1). A blank coefficient is 1.
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
Classify
0/21
Place each demand on the right construction.
Halving an angle
Sum of the sides
Perimeter and two angles
Perpendicular and midpoint
Explanation:
Halving is 11.1. The sum is 11.4. The perimeter is 11.6. The perpendicular is 11.2.
2
Place each difference in its case.
AB − AC
AC − AB
PR − PQ, angle at Q
AB − AC = 3.5 cm
Explanation:
If the first side is longer, use the ray. If the second is longer, use the opposite direction.
Very short answer
0/51
Write the two tools of a pure construction.
Model answer (board format)
An ungraduated straight edge and a compass.
Check your answer:
2
Write the formula and the value for 60°.
Model answer (board format)
180° ÷ 3 = 60°. The unit is the degree. The reason is an equilateral triangle.
Check your answer:
3
On AB = 8 cm write AM and the angle.
Model answer (board format)
AM = 8 ÷ 2 = 4 cm. The angle = 180° ÷ 2 = 90°.
Check your answer:
4
When does Construction 11.4 stop?
Model answer (board format)
When AB + AC ≤ BC.
Check your answer:
5
If the perimeter is 11 cm, how long is XY?
Model answer (board format)
XY = AB + BC + CA = 11 cm.
Check your answer:
Short answer
0/41
Obtain 15°, 45° and 75° from 60° and a right angle.
Model answer (board format)
15° = 60° ÷ 4. 45° = 90° ÷ 2. 75° = 60° + 15°. The unit is the degree.
Check your answer:
2
The base is 7 cm and the sum is 13 cm. How long is BD, and why is the construction possible?
Model answer (board format)
BD = 13 cm. 13 > 7, so AB + AC > BC. Construction 11.4 is possible.
Check your answer:
3
AB − AC = 3.5 cm and PR − PQ = 2 cm. Write the case and the direction.
Model answer (board format)
The first is case one, 3.5 cm on the ray. The second is case two, 2 cm in the opposite direction.
Check your answer:
4
Why is XB = AB in Construction 11.6? Also write XY for a perimeter of 12 cm.
Model answer (board format)
B lies on the perpendicular bisector of AX, so XB = AB. XY = 12 cm. The unit is the centimetre.
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Long answer
0/31
Write the reasons for Constructions 11.1 and 11.2. Then find half of 70° and half of 10 cm.
Model answer (board format)
In 11.1, SSS makes the angles equal. In 11.2, AM = BM and the linear pair gives 90°. 70° ÷ 2 = 35°. 10 ÷ 2 = 5 cm.
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2
Write the Exercise 11.1 angles 90°, 45°, 30°, 15°, 75°, 105° and 135° each as one sum or one division.
Model answer (board format)
90° = 60° + 30°. 45° = 90° ÷ 2. 30° = 60° ÷ 2. 15° = 30° ÷ 2. 75° = 60° + 15°. 105° = 60° + 45°. 135° = 90° + 45°.
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3
Which construction or case covers each of the five questions of Exercise 11.2? Do not change the order.
Model answer (board format)
1) Construction 11.4, sum 13 cm. 2) Construction 11.5 case one, 3.5 cm. 3) Construction 11.5 case two, 2 cm. 4) Construction 11.6, perimeter 11 cm. 5) The sum again, 18 cm, a right triangle.
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BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination. Annual questions will be added only when a source page is available.
1
22.5° equals —
💡 Hint:
A quarter of a right angle.
Explanation:
90° ÷ 4 = 22.5°.
2
Question 3 of Exercise 11.2 is —
💡 Hint:
PR − PQ.
Explanation:
Case two of the difference.
3
Half of 30° is ______ degrees.
💡 Hint:
30 ÷ 2.
Explanation:
15.
4
The first line of Construction 11.6 equals one side.
Explanation:
False — it equals the whole perimeter.
5
The base is 6 cm and AB + AC = 6 cm. Write the decision with the formula.
Model answer (board format)
The rule is AB + AC > BC. 6 = 6, so the construction is not possible. The unit is the centimetre.
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6
This is coefficient practice, not a result of constructions. Balance H2 + O2 making H2O by filling coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
CBSE-style questions
0/6These are competency-based practice questions. They are not copies of a CBSE paper.
1
A student draws 75° with a protractor and leaves the reason blank. A correct reason is —
💡 Hint:
Two known angles.
Explanation:
75° = 60° + 15°.
2
A corner on a map must be halved exactly. Which construction is used?
💡 Hint:
The bisector.
Explanation:
Construction 11.1.
3
Assertion (A): A pure construction uses a compass.
Reason (R): A protractor also takes the place of every step.
Explanation:
A is true. R is false — the protractor is a check, not a step.
4
If two sides and an angle which is not the included angle are given, the triangle is always unique.
Explanation:
False — the book says such a triangle is not always unique.
5
A window corner must be 90°, with a compass. Show the sum from 60°. Why is a protractor not enough?
Model answer (board format)
30° = 60° ÷ 2 and 90° = 60° + 30°. A protractor is only a check. The construction step is the compass. The unit is the degree.
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6
A triangular plot is said to have base 8 m and the sum of the other two sides 8 m. Why will an engineer stop the construction?
Model answer (board format)
AB + AC > BC is required. 8 m = 8 m, so the sum is not greater. Construction 11.4 is not possible.
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🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
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CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
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🧠 What you learned + equation sheet
What you learned
- A pure construction uses a straight edge and a compass.
- An angle bisector cuts the angle into two equal parts.
- A perpendicular bisector makes 90° at the midpoint.
- The 60° construction is an equilateral triangle. 30°, 15°, 90° and 45° are built from it.
- A triangle can be built from the base, a base angle, and the sum or the difference of the other two sides.
- Given the perimeter and two base angles, draw a line of that perimeter and bisect the angles.
| What | Keep this |
|---|---|
| Tools | straight edge + compass |
| Bisector | equal halves |
| Perp. bisector | 90° at midpoint |
| 60° | 180° ÷ 3 |
| Sum | AB + AC > base |
| Perimeter | XY = AB + BC + CA |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.