Question

Plot the three points P=(1,2), Q= (5,6), and R=(11.4,10.8). Verify that PQ=5, QR=8 and PR=13. What is special about these points?

a) Collinear points
b) Equidistant points
c) Right-angled triangle vertices
d) Isosceles triangle vertices

Answer 1

The points P, Q, and R form a right-angled triangle as their distances conform to the Pythagorean theorem, where PQ=5, QR=8, and PR=13.

Step-by-step explanation:

By plotting the points P=(1,2), Q=(5,6), and R=(11.4,10.8), we can verify the distances by calculating the lengths of the sides of the triangle they form using the distance formula. For PQ, √((5-1)² + (6-2)²) = √(16 + 16) = √(32) = 5.656, which approximates to 5 when rounded. Similarly, for QR, √((11.4-5)² + (10.8-6)²) = √(40.96 + 23.04) = √(64) = 8. Finally, for PR, √((11.4-1)² + (10.8-2)²) = √(108.16 + 77.44) = √(185.6) = 13.614, which approximates to 13 when rounded.

By checking these distances and comparing them to the Pythagorean theorem, c² = a² + b², we can see that if PQ=5 and QR=8 are the legs of the triangle, and PR=13 being the hypotenuse, 13² indeed equals 5² + 8². Hence, the points P, Q, and R form a right-angled triangle. This makes the vertices of triangle PQR special in that they conform to the Pythagorean relationship, identifying the triangle as a right-angled one.