Class 9 Mathematics

Chapter 12 Solutions — Quadrilaterals

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Solutions

Overview

Step-by-step NCERT solutions for Quadrilaterals (Chapter 12, NCERT Class 9 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Quadrilaterals textbook chapter.

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All 40 questions in Quadrilaterals are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise Set 12.1

1.
Let ABCD be a quadrilateral.
- (i) List all sides of ABCD adjacent to side AB. List all sides opposite to AB.
- (ii) List all angles of ABCD adjacent to ∠A. List all angles opposite to ∠A.
- (iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.
(*) You have used internal angles of quadrilaterals, but they too require an exact definition. Precisely define the internal angle of a quadrilateral at a given vertex. Your answer should work for a non-convex quadrilateral too. (Hint: use the opposite vertex as well.)
In a quadrilateral ABCD, suppose AB ‖ DC. Can ABCD be non-convex? What if we instead assume AB = CD? What if we instead assume ∠A = ∠C?
Consider three non-collinear points A, B, C and draw the lines AB, BC, CA. For every possible location of point D in the plane outside these lines, decide if ABCD is self-intersecting, non-convex, or convex. (Hint: the three lines divide the plane into 7 regions.)
Can a quadrilateral be both self-intersecting and non-planar?

Exercise Set 12.2

1.
True or false?
- (i) A parallelogram with a right angle is a rectangle.
- (ii) A rhombus with perpendicular diagonals is a square.
- (iii) If the diagonals of a parallelogram are equal, then it is a rectangle.
The diagonal AC of a parallelogram ABCD bisects ∠A. Show that it also bisects ∠C and that ABCD is a rhombus.
3.
The following questions examine converses of true properties. Answer them with Yes or No. If your answer is No, what extra condition can you add so that the answer becomes Yes?
- (i) If the diagonals of a quadrilateral ABCD bisect its angles, must ABCD be a rhombus?
- (ii) If the diagonals of a quadrilateral ABCD bisect each other at right angles, must ABCD be a rhombus?
- (iii) If the diagonals of a…
Let ABCD be a parallelogram with AB ≠ BC. Show that the pairwise intersection points of the four angle bisectors form the vertices of a rectangle (see Fig. 12.11). Why did we assume AB ≠ BC?

Exercise Set 12.3

1.
In ∆ABC, P, Q, R are the midpoints of sides AB, AC, BC respectively.
- (i) Show that ∆PQR is congruent to ∆QPA and to two other triangles which you should identify.
- (ii) Suppose someone erases ∆ABC, leaving only ∆PQR on the paper. Can you reconstruct ∆ABC from ∆PQR?
In ∆ABC, let M and N be midpoints of AB and AC respectively. Let D be any point on BC. Show that MN bisects AD.
In a quadrilateral ABCD, suppose AB ‖ DC and AB ≠ CD. Suppose G and H are the midpoints of AC and BD respectively. Prove that GH ‖ AB. (Why did we assume AB ≠ CD?)
4.
Suppose the midpoints of sides AB, BC, CD and DA of a quadrilateral ABCD are P, Q, R and S respectively.
- (i) Show that PR and QS bisect each other.
- (ii) Show that if AC = BD, then PR and QS are perpendicular. Is the converse true?
5.
Suppose PQRS is the Varignon parallelogram of ABCD.
- (i) Copy only PQRS on another paper. Show how you will recreate a congruent copy A′B′C′D′ of ABCD from PQRS.
- (ii) Justify why the quadrilateral A′B′C′D′ you constructed is congruent to ABCD. You may need to show why S is collinear with points A′ and D′ that you constructed, and similarly for P, Q and R. (Hint: Use congruence of triangles, for…

Exercise Set 12.4

Justify why the plane cannot be tiled with a regular pentagon. (Hint: Read the first 3 sentences of 'Think and Reflect' in the section on tiling.)
Draw a non-convex 4-gon DART. Show how we can tile the plane with copies of DART. Both methods that we discussed earlier will work. Which do you prefer?

End-of-Chapter Exercises

Using a fact about parallelograms, show how to tile the plane using any given triangle. (Hint: Can you use the parallelogram tiling in the introduction?)
Mark the midpoint of the line drawn on the paper (see Fig. 12.33), given that the horizontal lines are equally spaced. Justify your answer. [The figure shows a diagonal line segment crossing several equally spaced horizontal lines.]
You know that the sum of angles of a quadrilateral is 360°, even for a non-convex quadrilateral. Now consider a self-intersecting quadrilateral ABCD, where AB and CD intersect at point E. Show that ∠A + ∠B + ∠C + ∠D < 360°. Can you construct ABCD such that ∠A + ∠B + ∠C + ∠D = 2°?
In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 12.34). Show that APCQ is a parallelogram.
A right-triangle shaped cutout of a paper is folded such that point A touches point B (Fig. 12.35). Show that the crease line can be used to find the midpoint of not only AB but also that of AC.
You saw how to use the Midpoint Theorem to divide a given triangle into 4 congruent triangles. Can we divide a triangle into 3 congruent triangles? (i) Draw medians AP, BQ and CR of ∆ABC, meeting in a common point M. Cut along each median to get 6 triangles. Show with justification how to assemble the 6 pieces into 3 congruent triangles. Prove that you get a triangle. What are its side lengths?…
7.
A more general midpoint theorem and its converse. In a quadrilateral ABCD, suppose AB ‖ DC. (ABCD is called a trapezium.) Let E be the midpoint of AD. A line drawn through E intersects side BC at F.
- (i) If EF ‖ AB, then show that F is the midpoint of BC and conclude that EF = (AB + CD)/2.
- (ii) If F is the midpoint of BC, then show that EF ‖ AB.
The diagonals AC and BD of a parallelogram ABCD intersect at O. A line through O meets AB and CD at points P and Q respectively. Show that O is the midpoint of PQ. (Multiple proofs are possible. Which is the simplest?)
(*) ABCD is a trapezium with parallel sides AD = 3 cm and BC = 5 cm. E and F are the midpoints of the non-parallel sides. Find the ratio of the areas of the 4-gons AEFD and EBCF.
10.
Consider 4 points A, B, C, D in the plane with no three collinear.
- (i) How many different quadrilaterals do they form if the quadrilateral is allowed to be self-intersecting or non-convex?
- (ii) How many of these quadrilaterals are self-intersecting? How many are convex?
11.
Suppose P is a point on side AB of ∆ABC and the line through P parallel to BC meets AC in point Q. For parts
- (i) to (iii), assume AP = 1, AQ = √5, and see Fig. 12.39.
- (i) If PB = 2, find QC. (ii)* If PB = 1/3, find QC. (iii)* If PB = 2/3, find QC. (iv)* Show that if AP/PB is a rational number, then AP/PB = AQ/QC.
Suppose ABCD is a parallelogram and M, N are midpoints of AB and CD respectively. Show that segments DM and BN trisect segment AC. (ii) Use part (i) to find a procedure to trisect any given segment PQ. Find two other ways to trisect PQ, one using Exercise 11 above and a third using the Centroid Theorem.
13.
Is the Midpoint Theorem for Quadrilaterals (Theorem 9) true when the quadrilateral is non-convex? How about when it is self-intersecting?
- (i) Can you explain why this is so? See the geometric reasoning in the text.
- (ii) There is one exception: In a very special case the Varignon parallelogram becomes a single segment. When will this happen? *(iii) Does your reasoning apply even when ABCD is…
Let P, Q, R, S be four points on sides AB, BC, CD and DA respectively of a quadrilateral ABCD. Suppose PQRS is a parallelogram. Must P, Q, R and S be midpoints of the respective sides?
Complete the following proof of the Midpoint Theorem. Extend segment PQ beyond Q until point S. By how much should we extend PQ? (Hint: prove ∆APQ ≅ ∆CSQ.) (ii) Give a similar proof of the converse of the Midpoint Theorem (Theorem 7). Start by extending PQ up to a suitable point S.
(*) Review all the properties of a rhombus/rectangle/square that you proved in Grade 8. Formulate a converse of each. Decide if the converse is true. There are many possibilities here!
(*) Show that the sum of angles of a non-planar quadrilateral is always less than 360°. Can you find a non-planar quadrilateral ABCD for which ∠A + ∠B + ∠C + ∠D = 2°? (Hint: Think of a diagonal, say AC, as a hinge around which triangles ABC and ADC can rotate.) What happens to each angle of ABCD as you do this rotation?
18.
(*) A third method to tile the plane using a 4-gon. Two copies of SOME sharing a vertex appear to be shifted copies of each other.
- (i) Prove the exact match of SOME with S′O′M′E′ using four parallelograms.
- (ii) Follow the described procedure along each diagonal to build a 3×3 grid of 9 copies.
19.
(*) Is there a 4-gon with given side lengths?
- (i) Recall the triangle inequality: for a triangle with sides a, b, c (with a ≤ b ≤ c), the triangle exists iff a + b > c.
- (ii) Suppose a 4-gon has 2, 5, 11 as three side lengths. Can the fourth side be 100? Can it be 10? Can it be 1? What are the possible lengths of the fourth side?
- (iii) For given positive numbers a, b, c, d, how will you decide if…
20.
(*) Counting diagonals of a polygon.
- (i) How should we define a diagonal of an n-gon? How many diagonals does an n-gon have? Make a table for small values of n.
- (ii) Can you guess a formula for the number of diagonals? How many diagonals get added when we increase the number of sides by 1? Justify why your formula is true for all n.
(*) Sum of angles of a polygon. What is the sum of angles of a (planar non-self-intersecting) n-gon? We know the answer is 180° for n = 3 and 360° for n = 4. Find the next few values. Then find a formula in terms of n and prove it.
22.
(*) Multiple converses to a theorem. [See the textbook for the full statement with four named conditions: (P MID) P is midpoint of AB, (Q MID) Q is midpoint of AC, (PRLL) PQ ‖ BC, (HALF) PQ = BC/2.]
- (i) …
- (ii) Write the converse of the Midpoint Theorem in the form that gives Theorem 7. Write Theorem 7 in terms of the four named conditions.
- (iii) Examine the remaining combination: 'If (P MID) and (HALF) are true, can we conclude (Q MID) and/or (PRLL)?'
(*) Validity of tiling methods. Show using reasoning that each of the three tiling methods we saw produces a tiling of the plane using the given 4-gon. Prove that when we place new copies using any of the three procedures, the entire plane is covered with no gaps and no overlaps.
(*) What fraction of the square is shaded? [Fig. 12.43 shows a square with an inner rotated square (shaded) whose vertices lie at the midpoints of the outer square's sides. Equal tick marks confirm the midpoint division.]

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