Question

(Use a ruler and a compass for this question.)

1. Construct a triangle ABC such that BC = 8 cm, AC = 10 cm and ∠ABC = 90°.
2. Construct an incircle to this triangle. Mark the centre as I.
3. Measure and write the length of the in-radius.
4. Measure and write the length of the tangents from vertex C to the incircle.
5. Mark points P, Q and R where the in circle touches the sides AB, BC and AC of the triangle respectively. Write the relationship between ∠RIQ and ∠QCR.

Answer 1

Given:

Triangle ABC with BC = 8 cm, AC = 10 cm and ∠ABC = 90°, so B is the right angle.

Construct the incircle with centre I, measure the in‑radius and the tangents from C, mark touch points P, Q, R on AB, BC, AC and compare ∠RIQ and ∠QCR.

Step-wise calculation:

1. Find AB so you know all three sides before constructing:

AB² + BC² = AC²   ...(Pythagoras, since ∠B = 90°)

AB² = 10² – 8²

= 100 – 64

= 36

⇒ AB = 6 cm.

2. Construction ruler and compass, one convenient method:

1. Draw BC = 8 cm.
2. At B construct a line perpendicular to BC.
3. On that perpendicular mark A so that AB = 6 cm or intersect the perpendicular with the circle centered at C radius 10 cm either way gives A with AC = 10 and AB ⟂ BC.
4. Join A to C. Triangle ABC with sides 6, 8, 10 is obtained.

3. Construct the incircle and centre I:

1. Construct internal bisectors of any two angles for example ∠ABC and ∠BCA. Their intersection is the incenter I.
2. From I drop a perpendicular to any side say to BC, call the foot N; IN is the in‑radius.

4. Compute the in‑radius r exactly:

Semiperimeter `s = (AB + BC + AC)/2`

= `(6 + 8 + 10)/2`

= 12

Area = `1/2 xx AB xx BC`

= 0.5 × 6 × 8 

= 24

Inradius `r = "Area"/s`

= `24/12`

= 2 cm

So, IN = 2 cm.

5. Tangent length from vertex C to the incircle:

Standard tangent‑segment lengths: if the incircle touches AB at P, BC at Q and CA at R, then the two tangents from C (CQ and CR) are equal and each equals s – c, where c = AB the side opposite C.

Here, s – c

= 12 – 6

= 6 cm

So, each tangent from C to the incircle has length 6 cm.

6. Mark P, Q, R and compare angles:

Let P be touchpoint on AB, Q on BC, R on AC. By construction IR ⟂ AC and IQ ⟂ BC radius perpendicular to tangent.

∠RIQ is the angle between IR and IQ, i.e. the angle between the perpendiculars to AC and BC. The angle between two lines equals the angle between their perpendiculars, hence ∠RIQ = angle between AC and BC = ∠QCR.

Therefore, ∠RIQ = ∠QCR.