I’m Up and Down, and Round and Round Class 9 (Ganita Manjari I)

Draw △ABC with AB=5" " cm,∠" " A=70^∘ and ∠B=60^∘. Draw the circumcircle of △ABC. Is the centre inside or outside the triangle?
Since side AB is given, we follow these steps
Draw side AB
Make ∠" " 𝐀=〖𝟕𝟎〗^∘ and ∠𝐁=〖𝟔𝟎〗^∘
To make circumcircle, we construct perpendicular bisectors of two sides
Where the bisectors meet is center of circle, let’s mark it as O
Make a circle, taking center as O, radius as OA (or OB, OC since OA = OB = OC)

Draw △ABC with AB=5" " cm,∠" " A=100^∘,AC=4" " cm. Draw the circumcircle of △ABC. Is the centre inside or outside the triangle?
We follow these steps
Draw side AB
Make ∠" " 𝐀=〖𝟏𝟎𝟎〗^∘
In this ray, we mark point C such that AC = 4 cm
To make circumcircle, we construct perpendicular bisectors of two sides
Where the bisectors meet is center of circle, let’s mark it as O
Make a circle, taking center as O, radius as OA (or OB, OC since OA = OB = OC)

Draw △ABC, with AB=6" " cm,BC=7" " cm and CA=7" " cm. Draw the circumcircle of △ABC. Let the circumcentre be O . Measure OA, OB,OC.
We follow these steps
Draw side AB
Make AC = 7 cm from point A, BC = 7 cm from point B
Where they intersect is point C
To make circumcircle, we construct perpendicular bisectors of two sides
Where the bisectors meet is center of circle, let’s mark it as O
Make a circle, taking center as O, radius as OA (or OB, OC since OA = OB = OC)
Let’s do it step by step
Steps of construction – For making Triangle

What is the least possible radius of a circle through two points A and 𝐵?

Show that the triangle formed by a chord and the centre of the circle is isosceles.
Let’s make chord AB on the circle
Triangle formed by chord and center of circle is ∆ AOB

Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.
Let’s consider same circle and ∆ AOB from previous question

Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord? (Hint: Use Fig. 5.12. You are told that ∠CMA=∠CMB=90^∘. You need to show that AM=BM.)
Let’s do the proof of the converse

An isosceles triangle ABC is inscribed in a circle, with AB=AC. Show that the altitude from A to BC passes through the centre of the circle.
Given: ∆ ABC inscribed in circle
With AB = AC

Two parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. If the radius of the circle is 5 cm, find the distance between the midpoints of the chords.
Let’s draw the figure

Use the Baudhāyana–Pythagoras theorem to show why Theorem
6 must be true.
Theorem 6 is
Chords of a circle having the same length are all at the same distance from the centre of the circle

Consider Fig. 5.15. If CE is perpendicular to AB,CH is perpendicular to GH, and CE =CH, show that AB=GF.
Given: : 𝐶𝐸⊥𝐴𝐵 and 𝐶𝐻⊥𝐺𝐹.
And, distances are equal i.e. CE = CH

Solve the previous question using the Baudhāyana-Pythagoras theorem.
Given: : 𝐶𝐸⊥𝐴𝐵 and 𝐶𝐻⊥𝐺𝐹.
And, distances are equal i.e. CE = CH

Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance is 6 cm.
Let’s draw the diagram
Here, we have circle with center O
And, Radius = 7 cm
Let AB be the chord
And, OM be perpendicular distance to AB from O
∴ OM = 6 cm

Explain why the following statement is true: If the perpendicular distance of a chord from the centre is 𝑑 and the radius is 𝑟, then the chord length is 2√(𝑟^2−𝑑^2 ).
This is the same as previous question, but without the values

In a circle, if the distance of chord AB from the centre is twice the distance of another chord CD from the centre, then can we conclude that CD=2AB? Give reasons for your answer.
We draw a diagram
Let Radius = r

In a circle with centre O, the central angle AOB is 60^∘. If the radius of the circle is 12 cm, what is the length of the chord AB?
Let’s draw a figure

Let A and B be two points on a circle with centre O. (i) Are there points X,Y on the circle, on the same side of AB, such that ∠AXB is different from ∠AYB?
(ii) Is it true that if ∠AXB=∠AYB, then X and Y lie on the same side of the circle?
(iii) If ∠AXB=∠AYB, and X and Y do not lie on the circle, does the circle through A,B and X also pass through Y?

(Method 1) Find 𝑥 in Fig. 5.26.
Here, we will use the property
Opposite angles of Cyclic Quadrilateral are supplementary

There is a unique circle passing through three non-collinear points

Equal chords of a circle subtend equal angles at the centre of the circle

Chords of a circle that subtend equal angles at the centre are equal.

The line joining the centre of a circle and the midpoint of
a chord of the circle is perpendicular to the chord

The perpendicular from the centre of a circle to a chord of the circle bisects the chord

Chords of a circle having the same length are all at the same distance from the centre of the circle

Chords of a circle that are equidistant from the centre have equal length

Let AB and DE be two chords of a circle with centre C. Suppose AB > DE. Then the distance from C to AB is less than the distance from C to DE

The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc at any point on the circle outside the arc.

Theorem
Angle subtended by a diameter/semicircle on any point of circle is 90°

Theorem
Angles in the same segment of a circle are equal.

If a line segment AB joining two points A, B subtends equal angles at two other points C, D that lie on the same side of AB, then the four points lie on a circle.

The sum of two opposite angles of a cyclic quadrilateral is 180°.

If two opposite angles of a quadrilateral add up to 180°, then the vertices of the quadrilateral lie on a circle, i.e., they are concyclic.

In a circle, a chord is 5 cm away from the centre. If the radius of the circle is 13 cm, what is the length of the chord?
Let’s draw the diagram
Here, we have circle with center O
And, Radius = 13 cm
Let AB be the chord
And, OM be perpendicular distance to AB from O
∴ OM = 5 cm

An arc of a circle subtends an angle of 70^∘ at the centre. What is the measure of the angle subtended by the arc at a point on the circle?
Drawing figure
Let arc AB subtend ∠ AOB = 70° at the center

The diameter of a circle is 26 cm. A chord of length 24 cm is drawn in the circle. Find the distance from the centre of the circle to the chord.
Let’s draw the diagram
Here, we have circle with center O
Since Diamter is 26 cm
Radius = 26/2 cm = 13 cm

A circle has a radius of 15 cm. A chord is drawn. The distance from the centre of the circle to the chord is 9 cm. What is the length of the chord?
Let’s draw the diagram
Here, we have circle with center O
And, Radius = 15 cm
Let AB be the chord
And, OM be perpendicular distance to AB from O
∴ OM = 9 cm

Prove that the perpendicular bisector of a chord passes through the centre of the circle.
Given: AB is a chord
And PM is perpendicular bisector of AB
∴ PM ⊥ AB
& M is mid-point of AB, i.e. AM = BM

The diameter of a circle is AB . Point C is on the circumference. What is the measure of the ∠ACB? Explain your reasoning.
We know that
Angle in a semicircle is always a right angle

ABCD is a cyclic quadrilateral inscribed in a circle. If ∠A measures 75^∘, what is the measure of ∠C? If ∠B measures 110^∘, what is the measure of ∠D ?
In a cyclic quadrilateral
Opposite angles are supplementary

Quadrilateral PQRS is inscribed in a circle. If ∠P=(2𝑥+10)^∘ and ∠R=(3𝑥−20)^∘, find the value of 𝑥 and the measures of ∠P and ∠R.
In a cyclic quadrilateral
Opposite angles are supplementary

The distance of a chord of length 16 cm from the centre of a circle is 6 cm. Find the radius of the circle.
Let’s draw the diagram
Here, we have circle with center O
Let Radius = r cm

A cyclic quadrilateral has sides 5, 5, 12, 12 units Find its area.
Depending on the order of the sides, this shape is either
a Rectangle (5, 12, 5, 12)
or a Kite (5, 5, 12, 12)

Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out?
Our property is
Centre of any circumcircle (the circumcentre) is always located where the perpendicular bisectors of the shape's sides intersect

When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord.
Given: Let AB & CD be the two equal chords
intersecting at point X.
∴ AB = CD

Draw a circle in which a chord of 6 cm length stands at a distance of 3 cm from the centre.
(Hint: Is it a circumcircle of a suitable triangle?)
Let’s draw a rough diagram
Here, radius of circle and center is not given

Show that rectangle is the only parallelogram that can be inscribed in a circle.
Given: Let ABCD be a cyclic parallelogram

Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.
Given: A circle with center O
ABCD is a rectangle inscribed in a circle
with diagonals AC & BD

Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?
Let’s do this step-by-step
Step 1 of 6
The Single Chord
Let's start with an outer circle of radius
. We draw a single chord of a fixed length
.
The green dot marks the exact midpoint of this chord.
Previous
Next StepStep 2 of 6
2. Multiple Chords
Now, imagine drawing many different chords around the circle, all with the exact same length
.
We mark the midpoint of each one. Notice a pattern starting to emerge?
Previous
Next StepStep 3 of 6
3. The Locus (The Answer)
If we draw every possible chord of length L, their midpoints form a continuous path.

In a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: "The centre of the circle lies on the angle bisector of ∠BAC ".
Given: A circle with center O. Two equal chords, AB and AC
i.e. AB = AC.

Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm . Find the radius of the circle.
Let’s draw the figure

A regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.
We have a regular hexagon inscribed in a circle with center O and radius r.

A quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP? Explain your reasoning.
Let’s draw the figure

Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., ∠CDE=∠ABC, where E is a point on the extension of side CD).
Let’s draw the figure

There is no chord of a circle that is longer than its diameter." How do you justify this statement?
Let’s do this step-by-step
Step 1 of 5
The Diameter
Let's start with a circle with center
and a radius of
.
The diameter is a special chord that passes straight through the center. Its total length is exactly
.
Previous
Next StepStep 2 of 5
2. A Random Chord
Now, let's draw any other random chord, AB, that does not pass through the center.
We want to prove that this red chord AB must be strictly shorter than the blue diameter.
Previous
Next StepStep 3 of 5
3. Forming a Triangle
Draw a line from the center
to point
, and another from
to
.
These are both radii of the circle, so their lengths are both exactly
.
Notice that we have now formed a triangle:
.
Previous
Next StepStep 4 of 5
4. The Triangle Inequality
A fundamental rule of geometry is the Triangle Inequality Theorem: The shortest distance between two points is a straight line.
This means walking from A to B directly (the chord) is always shorter than taking the detour from A to O , and then O to B .
Previous
Next StepStep 5 of 5
5. The Final Justification
Let's write this out mathematically:
Distance
Distance

Let A be any point within a given circle with centre O. Show that the shortest chord of the circle that passes through point A is the one that is perpendicular to OA.

How would you use the following figure to justify the statement that the angle in a semicircle is 90^∘?
Let’s label the points at the end as M & N
We need to prove
∠ MAN = 90°

In a circle, two chords CC^′ and DD ' are drawn perpendicular to a diameter AB. Prove that the segment MM ' joining the midpoints of the chords CD and C^′ D^′ is perpendicular to AB.
We use the concept of reflectional symmetry across the diameter to prove this.

How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is 180^∘?
Here,
Each small triangle (eg: ∆ OBC) is an isosceles triangle
And, angles opposite to equal sides are equal
So, ∠ OCB = ∠ OBC = q
Similarly, we can find all angles
And, at the end we can use Angle sum property of Quadrilateral to prove opposite angles are supplementary