Introduction / Context:
This question tests knowledge of standard formulas for special triangles, in particular an equilateral triangle. The inradius of a triangle is the radius of the circle that touches all three sides. For an equilateral triangle, there are well known formulas connecting side length, height, area, circumradius and inradius.
Given Data / Assumptions:
- PQR is an equilateral triangle.
- Each side has length a = 10 cm.
- We are asked to find the inradius r of this triangle.
- The triangle is regular, so all angles are 60°, and standard formulas for equilateral triangles apply.
Concept / Approach:
For an equilateral triangle of side a, the height h is given by h = (√3 / 2) * a. The inradius r of any triangle is related to its area K and semi perimeter s by r = K / s. In an equilateral triangle, we can also use the direct formula r = a√3 / 6, which is derived from the height relation r = h / 3. We will use this simpler direct formula to compute the inradius for side 10 cm.
Step-by-Step Solution:
1. For an equilateral triangle of side a, the height h is h = (√3 / 2) * a.
2. Here a = 10 cm, so h = (√3 / 2) * 10 = 5√3 cm.
3. The inradius r for an equilateral triangle is given by r = h / 3.
4. Substitute h = 5√3: r = (5√3) / 3 cm.
5. Another standard formula is r = a√3 / 6. For a = 10, r = 10√3 / 6 = (5√3) / 3, which matches the previous result.
6. To express r in the form shown in the options, note that 5/√3 is equivalent to (5√3) / 3 after rationalising the denominator.
7. Therefore r = 5√3 / 3 cm = 5 / √3 cm, which matches option A.
Verification / Alternative check:
We can cross check using the area formula. The area of an equilateral triangle is K = (√3 / 4) a^2. For a = 10, K = (√3 / 4) * 100 = 25√3. The semi perimeter is s = (3a) / 2 = 15. Then the inradius is r = K / s = (25√3) / 15 = (5√3) / 3, which again equals 5 / √3 cm after rationalisation. This confirms the result via a different method.
Why Other Options Are Wrong:
Option B (10√3) is far too large, even larger than the side length, which is impossible for an inradius inside the triangle.
Option C (10/√3) corresponds to twice the inradius and would be larger than the height of the triangle.
Option D (5√2) does not align with any standard formula for equilateral triangles and evaluates to a value incompatible with the geometry of side 10 cm.
Common Pitfalls:
Learners sometimes confuse the inradius formula with the circumradius formula R = a√3 / 3, or they mistakenly use the height h in place of r. Another common slip is forgetting to divide by three when relating the inradius to the height. Knowing and remembering that r = a√3 / 6 for an equilateral triangle helps avoid these issues.
Final Answer:
The inradius of triangle PQR is
5/√3 cm.
Discussion & Comments