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

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 $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. We substitute the coordinates of $P$ and $Q$ into this formula.

Step 3: Substitute values for PQ

Substituting the coordinates $x_1 = 3$, $y_1 = 2$ and $x_2 = -2$, $y_2 = -3$ into the distance formula, we get $PQ = \sqrt{(-5)^2 + (-5)^2}$ which simplifies to $PQ = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$ 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 $x_1 = -2$, $y_1 = -3$ and $x_2 = 2$, $y_2 = 3$ into the distance formula, we get $QR = \sqrt{(4)^2 + (6)^2}$ which simplifies to $QR = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$ 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 $x_1 = 3$, $y_1 = 2$ and $x_2 = 2$, $y_2 = 3$ into the distance formula, we get $PR = \sqrt{(-1)^2 + (1)^2}$ which simplifies to $PR = \sqrt{1 + 1} = \sqrt{2}$ 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: $PQ = 5\sqrt{2} \approx 7.07$, $QR = 2\sqrt{13} \approx 7.21$, and $PR = \sqrt{2} \approx 1.41$. 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: $PQ^2 = (5\sqrt{2})^2 = 50$, $QR^2 = (2\sqrt{13})^2 = 52$, and $PR^2 = (\sqrt{2})^2 = 2$. 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: $PQ^2 + PR^2 = 50 + 2 = 52$. This sum is equal to the square of the longest side, $QR^2 = 52$. Thus, $PQ^2 + PR^2 = QR^2$ 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.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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