StemCET Logo

Maths Question 9 – JEE-MAIN 2026

← Back
Let the mid points of the sides of a triangle ABC be (52,7), (52,3) and (4,5). If its incentre is (h,k), then 3h+k is equal to :

The sum of the coordinates of the vertices of a triangle is twice the sum of the coordinates of the midpoints of its sides.

Step 1: Determine the Vertices of the Triangle✦ Active

Let the vertices be A(x1,y1), B(x2,y2), C(x3,y3) and the midpoints be D(52,7), E(52,3), F(4,5). Using the midpoint formula, we have:

x1+x2=5,x2+x3=5,x3+x1=8

Summing these gives 2(x1+x2+x3)=18⟹x1+x2+x3=9. Solving for xi: x1=9−5=4, x2=9−8=1, x3=9−5=4. Similarly for y-coordinates:

y1+y2=14,y2+y3=6,y3+y1=10

Summing these gives 2(y1+y2+y3)=30⟹y1+y2+y3=15. Solving for yi: y1=15−6=9, y2=15−10=5, y3=15−14=1. Thus, the vertices are A(4,9), B(1,5), and C(4,1).

Step 2: Calculate Side Lengths and Incenter Coordinates○ Expand

Calculate the side lengths using the distance formula:

a=BC=(4−1)2+(1−5)2=32+(−4)2=9+16=5
b=AC=(4−4)2+(1−9)2=02+(−8)2=64=8
c=AB=(1−4)2+(5−9)2=(−3)2+(−4)2=9+16=5

The incenter (h,k) is given by the formula:

h=ax1+bx2+cx3a+b+c=5(4)+8(1)+5(4)5+8+5=20+8+2018=4818=83
k=ay1+by2+cy3a+b+c=5(9)+8(5)+5(1)5+8+5=45+40+518=9018=5

So, the incenter is (h,k)=(83,5).

Step 3: Compute the Final Expression○ Expand

The required value is 3h+k:

3h+k=3(83)+5=8+5=13
✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.