To show that the points A(8,3), B(0,9), and C(14,11) are the vertices of an isosceles right-angled triangle, we will follow these steps:
### Step 1: Calculate the lengths of the sides using the distance formula.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
#### 1.1 Calculate AB:
For points A(8,3) and B(0,9):
\[
AB = \sqrt{(0 - 8)^2 + (9 - 3)^2}
\]
\[
= \sqrt{(-8)^2 + (6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]
#### 1.2 Calculate BC:
For points B(0,9) and C(14,11):
\[
BC = \sqrt{(14 - 0)^2 + (11 - 9)^2}
\]
\[
= \sqrt{(14)^2 + (2)^2}
\]
\[
= \sqrt{196 + 4}
\]
\[
= \sqrt{200}
\]
\[
= 10\sqrt{2}
\]
#### 1.3 Calculate AC:
For points A(8,3) and C(14,11):
\[
AC = \sqrt{(14 - 8)^2 + (11 - 3)^2}
\]
\[
= \sqrt{(6)^2 + (8)^2}
\]
\[
= \sqrt{36 + 64}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]
### Step 2: Verify if the triangle is isosceles.
From our calculations, we have:
- \(AB = 10\)
- \(BC = 10\sqrt{2}\)
- \(AC = 10\)
Since \(AB = AC\), the triangle is isosceles.
### Step 3: Verify if the triangle is right-angled.
To check if it is a right-angled triangle, we will use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Assuming \(BC\) is the hypotenuse:
\[
BC^2 = AB^2 + AC^2
\]
Calculating:
\[
(10\sqrt{2})^2 = 10^2 + 10^2
\]
\[
200 = 100 + 100
\]
\[
200 = 200
\]
Since the equation holds true, the triangle is a right-angled triangle.
### Conclusion
The triangle formed by points A(8,3), B(0,9), and C(14,11) is an isosceles right-angled triangle.
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