NIMCET 2013 Mathematics PYQ — The equation of the base of an equilateral triangle is and one ve… | Mathem Solvex | Mathem Solvex
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NIMCET 2013 — Mathematics PYQ
NIMCET | Mathematics | 2013
The equation of the base of an equilateral triangle is x+y=2 and one vertex is (2,−1). The length of the side of the triangle is:
Choose the correct answer:
A.
23
B.
2
C.
32
(Correct Answer)
D.
320
Correct Answer:
32
Explanation
Solution
Concept:
In an equilateral triangle, the perpendicular distance from a vertex to the opposite side (the base) is the altitude (h). The relationship between the side length (a) and the altitude (h) is given by:
h=23a⟹a=32h
Step 1: Find the altitude (h).
The altitude is the perpendicular distance from the vertex (x1,y1)=(2,−1) to the line x+y−2=0.
The formula for the distance d from a point (x1,y1) to a line Ax+By+C=0 is:
d=A2+B2∣Ax1+By1+C∣
Applying the values:
h=12+12∣(1)(2)+(1)(−1)−2∣
h=2∣2−1−2∣
h=2∣−1∣=21
Step 2: Calculate the side length (a).
Substitute the value of h into the side-altitude relation:
a=32×h
a=32×21
Step 3: Simplify the expression.
We can write 2 as 2×2:
a=3×22×2
a=32
a=32
Final Answer:
The length of the side of the triangle is 32, which corresponds to Option 3.
Explanation
Solution
Concept:
In an equilateral triangle, the perpendicular distance from a vertex to the opposite side (the base) is the altitude (h). The relationship between the side length (a) and the altitude (h) is given by:
h=23a⟹a=32h
Step 1: Find the altitude (h).
The altitude is the perpendicular distance from the vertex (x1,y1)=(2,−1) to the line x+y−2=0.
The formula for the distance d from a point (x1,y1) to a line Ax+By+C=0 is:
d=A2+B2∣Ax1+By1+C∣
Applying the values:
h=12+12∣(1)(2)+(1)(−1)−2∣
h=2∣2−1−2∣
h=2∣−1∣=21
Step 2: Calculate the side length (a).
Substitute the value of h into the side-altitude relation:
a=32×h
a=32×21
Step 3: Simplify the expression.
We can write 2 as 2×2:
a=3×22×2
a=32
a=32
Final Answer:
The length of the side of the triangle is 32, which corresponds to Option 3.
NIMCET
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