Are the points A (3,6,9), B (10,20,30) and C(25,−41,5), the vertices of a right angled triangle?
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Step-by-Step Solution
Step 1: Calculate the square of the distance between each pair of points
To determine if the points form a right-angled triangle, we first need to calculate the square of the distance between each pair of points using the distance formula in 3D: d2=(x2−x1)2+(y2−y1)2+(z2−z1)2. We will calculate AB2, BC2, and AC2.
Step 2: Compute the squared distances
Now, we perform the arithmetic for each squared distance. For AB2, we get 49+196+441=686. For BC2, we get 225+3721+625=4571. For AC2, we get 484+2209+16=2709.
Step 3: Apply the Pythagorean theorem
For the points to form a right-angled triangle, the sum of the squares of the two shorter sides must be equal to the square of the longest side (Pythagorean theorem). We check all three possible combinations.
Step 4: Compare the sums of squares
Comparing the sums, we find that AB2+AC2=3395, which is not equal to BC2=4571. Similarly, AB2+BC2=5257, which is not equal to AC2=2709. And BC2+AC2=7280, which is not equal to AB2=686. Since none of the combinations satisfy the Pythagorean theorem, the given points do not form a right-angled triangle.