**1. Which of the following steps of construction is INCORRECT while constructing a circumscribing circle of an equilateral triangle ABC of side 5 cm?**

**Step 1: Draw a line segment AB = 5 cm.**

**Step 2: With A and B as the centres, draw arcs of radius 5 cm intersecting each other at C.**

**Step 3: Join AC and BC. △ABC is an equilateral triangle.**

**Step 4: Draw the median of AB and AC that intersect each other at O.**

**Step 5: Join OA.**

**Step 6: With O as the centre and OA as the radius, draw a circle passing through A, B and C.**

a) Step 2

b) Step 6

c) Step 4

d) Step 3

**Answer: **c) Step 4

**Explanation:** Consider the figure given below:

The steps to construct a circumscribing circle of an equilateral triangle ABC of side 5 cm are:

**Step 1: **Draw a line segment AB = 5 cm.

**Step 2: **With A and B as the centres, draw arcs of radius 5 cm intersecting each other at C.

**Step 3: **Join AC and BC. △ABC is an equilateral triangle.

**Step 4:** Draw the perpendicular bisectors of side AB and AC intersecting each other at O.

**Step 5: **Join OA.

**Step 6: **With O as the centre and OA as the radius, draw a circle passing through A, B and C.

**2. If the angle bisectors of angles P and R of a triangle PQR meet at O, then what is the point O called?**

a) Centroid

b) Orthocentre

c) Circumcentre

d) Incentre

**Answer: **d) Incentre

**Explanation:**

**Incentre:**The incentre of a triangle is where the three angle bisectors meet. It is like the centre of a circle that fits inside the triangle perfectly and touches all three sides.**Circumcentre:**The circumcentre is a point where the perpendicular bisectors of the three sides of a triangle intersect. It is the centre of a circle that passes through all three vertices of the triangle.**Orthocentre:**The orthocentre is a point where the three altitudes of a triangle intersect. It is where the lines drawn straight up from each corner meet.**Centroid:**The centroid is a point where the three medians (lines joining each vertex to the midpoint of the opposite side) of a triangle intersect.

**3. What is the measure of each internal angle of a regular hexagon?**

a) 100°

b) 110°

c) 120°

d) 130°

**Answer: **c) 120°

**Explanation:** The interior angle of the regular polygon of n sides is given by:

Each interior angle = $\frac{\frac{\mathrm{2n\; -\; 4}}{}}{\frac{n}{}}$ × 90°

A regular hexagon has 6 sides.

Thus, n = 6

Each interior angle of a regular hexagon = $\frac{\frac{\mathrm{2n\; -\; 4}}{}}{\frac{n}{}}$ × 90°

= $\frac{\frac{\mathrm{2(6)\; -\; 4}}{}}{\frac{6}{}}$ × 90°

= (12 − 4) × 15°

= 8 × 15°

= 120°

**Each interior angle of a regular hexagon = 120°**

**4. What will be the radius of the circle circumscribing an equilateral triangle of side 4.5 cm constructed using a ruler and compass?**

a) 2.4 cm

b) 2.6 cm

c) 3.4 cm

d) 3.6 cm

**Answer: **b) 2.6 cm

**Explanation:** Construct the figure using the given steps:

**Steps of Construction:**

Step 1: Make a line segment BC of length 4.5 cm.

Step 2: Use points B and C as centres and draw two arcs with a radius of 4.5 cm intersecting each other at point A.

Step 3: Join AC and AB.

Step 4: Draw the perpendicular bisectors of AC and BC intersecting each other at O.

Step 5: Using O as the centre and OA, OB or OC as the radius, draw a circle. This circle will pass through points A, B, and C.

Step 6: This is the required circumcircle of triangle ABC. After measuring, the radius is OA is 2.6 cm.

**5. Arrange the given steps for constructing an inscribing circle of a regular hexagon of side 6 cm in CORRECT order.**

**Steps of Construction:**

**Step 1: At points A and B, draw the angle bisectors of angles A and B intersecting each other at point O.**

a) 3 - 5 - 6 - 1 - 7 - 4 - 2

b) 3 - 1 - 6 - 5 - 7 - 4 - 2

c) 3 - 6 - 5 - 7 - 1 - 4 - 2

d) 3 - 1 - 5 - 6 - 7 - 4 - 2

**Answer: **c) 3 - 6 - 5 - 7 - 1 - 4 - 2

**Explanation: **

**Steps of Construction:**

Step 1: Draw a line segment AB of length 6 cm.

Step 2: Draw rays making an angle of 120° each at points A and B. These lines cut off segments AF and BC, each measuring 6 cm.

Step 3: At points F and C, draw rays making an angle of 120° each. These lines cut off segments FE and CD, both measuring 6 cm.

Step 4: Join DE. Thus, ABCDEF is the required regular hexagon.

Step 5: At A and B, draw the angle bisectors of angles A and B intersecting each other at point O.

Step 6: From O, draw OL perpendicular to AB.

Step 7: With O as the centre and OL as the radius, draw a circle touching the hexagon's sides. Thus, this circle is the required incircle.

Construct the figure using the given steps:

>> Join CREST Olympiads WhatsApp Channel for latest updates.

If your web browser doesn't have a PDF Plugin. Instead you can Click here to download the PDF

>> Join CREST Olympiads WhatsApp Channel for latest updates.

In this section, you will find interesting and well-explained topic-wise video summary of the topic, perfect for quick revision before your Olympiad exams.

×

>> Join CREST Olympiads WhatsApp Channel for latest updates.