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Mathematics · Coordinate Geometry · NCERT Examples

Example 1

A complete, independently verified solution for NCERT Example 1.

Mathematics · Chapter 7 · NCERT Examples

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NCERT Example

Example 1

distance formula · right triangle · converse of Pythagoras theorem

Do the points P(3, 2), Q(-2, -3) and R(2, 3) form a triangle? If so, name the type of triangle formed.

Solution

Board-exam working:

Using the distance formula:

PQ=(3−(−2))2+(2−(−3))2PQ=\sqrt{(3-(-2))^2+(2-(-3))^2}
PQ=52+52=50=52PQ=\sqrt{5^2+5^2}=\sqrt{50}=5\sqrt2
QR=(2−(−2))2+(3−(−3))2QR=\sqrt{(2-(-2))^2+(3-(-3))^2}
QR=42+62=52=213QR=\sqrt{4^2+6^2}=\sqrt{52}=2\sqrt{13}
PR=(3−2)2+(2−3)2PR=\sqrt{(3-2)^2+(2-3)^2}
PR=12+(−1)2=2PR=\sqrt{1^2+(-1)^2}=\sqrt2

The sum of any two distances is greater than the third, so the three points form a triangle.

Now:

PQ2+PR2=50+2=52=QR2PQ^2+PR^2=50+2=52=QR^2

By the converse of the Pythagoras theorem, ∠P = 90°. Therefore, △PQR is right-angled at P.

Final answer

The points form a right triangle, right-angled at P.