The coordinates of the three vertices of a triangle are (3, 4), (15, 4) and (3, 13). Then, the radius of the incircle of the triangle is:
Answer and solution
Correct answer: 3
Solution
Answer: 3
1. The triangle is right-angled because one side is horizontal and another is vertical.
2. The two perpendicular sides are:
3. 15 − 3 = 12
13 − 4 = 9
4. Using the 9-12-15 Pythagorean triplet, the hypotenuse is 15.
5. For a right-angled triangle, the inradius is:
6. (radius) = (sum of the two perpendicular sides − hypotenuse) ÷ 2
7. So,
8. r = (12 + 9 − 15) ÷ 2
r = 6 ÷ 2
r = 3
9. Answer: 3 (TITA).
Why this method works:
The vertices with common y-coordinate give the horizontal leg, and those with common x-coordinate give the vertical leg. Their meeting point is the right angle. Translation of the triangle does not change its side lengths or inradius.
Independent check and common trap:
An independent area check gives area=12×9/2=54 and semiperimeter=(12+9+15)/2=18. Since area=inradius×semiperimeter, r=54/18=3. This verifies the right-triangle shortcut.
Apply: Read the legs 9 and 12; hypotenuse 15. Inradius=(9+12−15)/2=3.
Check: An independent area check gives area=12×9/2=54 and semiperimeter=(12+9+15)/2=18. Since area=inradius×semiperimeter, r=54/18=3. This verifies the right-triangle shortcut.
1. The triangle is right-angled because one side is horizontal and another is vertical.
2. The two perpendicular sides are:
3. 15 − 3 = 12
13 − 4 = 9
4. Using the 9-12-15 Pythagorean triplet, the hypotenuse is 15.
5. For a right-angled triangle, the inradius is:
6. (radius) = (sum of the two perpendicular sides − hypotenuse) ÷ 2
7. So,
8. r = (12 + 9 − 15) ÷ 2
r = 6 ÷ 2
r = 3
9. Answer: 3 (TITA).
Why this method works:
The vertices with common y-coordinate give the horizontal leg, and those with common x-coordinate give the vertical leg. Their meeting point is the right angle. Translation of the triangle does not change its side lengths or inradius.
Independent check and common trap:
An independent area check gives area=12×9/2=54 and semiperimeter=(12+9+15)/2=18. Since area=inradius×semiperimeter, r=54/18=3. This verifies the right-triangle shortcut.
G Strategy shortcut
Spot: Geometry – Coordinate Geometry.Apply: Read the legs 9 and 12; hypotenuse 15. Inradius=(9+12−15)/2=3.
Check: An independent area check gives area=12×9/2=54 and semiperimeter=(12+9+15)/2=18. Since area=inradius×semiperimeter, r=54/18=3. This verifies the right-triangle shortcut.
G Strategy: Spot: Geometry – Coordinate Geometry. Apply: Read the legs 9 and 12; hypotenuse 15. Inradius=(9+12−15)/2=3. Check: An independent area check gives area=12×9/2=54 and semiperimeter=(12+9+15)/2=18. Since area=inradius×semiperimeter, r=54/18=3. This verifies the right-triangle shortcut.