Do the points (3,2), (−2,−3) and (2,3) form a triangle? If so, name the type of triangle formed.
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Step-by-Step Solution
Step 1: Define the points
We first label the given points as P, Q, and R for clarity in applying the distance formula.
Step 2: Calculate the distance PQ using the distance formula
We use the distance formula to find the length of the segment connecting points P(3,2) and Q(−2,−3). The distance formula is given by (x2−x1)2+(y2−y1)2. We substitute the coordinates of P and Q into this formula.
Step 3: Substitute values for PQ
Substituting the coordinates x1=3, y1=2 and x2=−2, y2=−3 into the distance formula, we get PQ=(−5)2+(−5)2 which simplifies to PQ=25+25=50=52 units.
Step 4: Calculate the distance QR using the distance formula
Next, we calculate the distance between points Q(−2,−3) and R(2,3). We apply the distance formula using their respective coordinates.
Step 5: Substitute values for QR
Substituting the coordinates x1=−2, y1=−3 and x2=2, y2=3 into the distance formula, we get QR=(4)2+(6)2 which simplifies to QR=16+36=52=213 units.
Step 6: Calculate the distance PR using the distance formula
Finally, we calculate the distance between points P(3,2) and R(2,3) using the distance formula.
Step 7: Substitute values for PR
Substituting the coordinates x1=3, y1=2 and x2=2, y2=3 into the distance formula, we get PR=(−1)2+(1)2 which simplifies to PR=1+1=2 units.
Step 8: Check triangle inequality theorem
For three points to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. We now check this condition with our calculated distances: PQ=52≈7.07, QR=213≈7.21, and PR=2≈1.41. The sums are PQ+PR=7.07+1.41=8.48>7.21(QR), PQ+QR=7.07+7.21=14.28>1.41(PR), and QR+PR=7.21+1.41=8.62>7.07(PQ). Since all conditions are met, the points form a triangle.
Step 9: Check for right triangle using Pythagorean theorem
To determine the type of triangle, we check if it's a right triangle using the converse of the Pythagorean theorem. We square the lengths of the sides: PQ2=(52)2=50, QR2=(213)2=52, and PR2=(2)2=2. We check if the sum of the squares of the two shorter sides equals the square of the longest side.
Step 10: Verify Pythagorean theorem
We add the squares of the two shorter sides: PQ2+PR2=50+2=52. This sum is equal to the square of the longest side, QR2=52. Thus, PQ2+PR2=QR2 is true. According to the converse of the Pythagorean theorem, the triangle PQR is a right-angled triangle with the right angle at P, the vertex opposite the hypotenuse QR.