Using the distance formula between two points (x_1, y_1) and (x_2, y_2), the squared distance is (x_2 - x_1)² + (y_2 - y_1)². Applying this formula gives AB² = (a - 1)² + 1, BC² = 1 + (1 - b)², and AC² = a² + b².

Using the distance formula between two points (x_1, y_1) and (x_2, y_2), the squared distance is (x_2 - x_1)² + (y_2 - y_1)². Applying this formula gives AB² = (a - 1)² + 1, BC² = 1 + (1 - b)², and AC² = a² + b².

We now substitute the expressions for AC², AB², and BC² directly into the equation AC² = AB² + BC². Expanding the quadratic expressions yields a² + b² = (a² - 2a + 1 + 1) + (1 + 1 - 2b + b²).

We now substitute the expressions for AC², AB², and BC² directly into the equation AC² = AB² + BC². Expanding the quadratic expressions yields a² + b² = (a² - 2a + 1 + 1) + (1 + 1 - 2b + b²).

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]

Answer: $a + b = 2$

Step-by-step solution

Step 1: Recall the Pythagoras theorem for a right-angled triangle

Since $\triangle ABC$ is right-angled at vertex $B$ , the side opposite to $\angle B$ , which is $AC$ , forms the hypotenuse. According to the Pythagoras theorem, the square of the hypotenuse equals the sum of the squares of the other two sides.

Step 2: Compute the squared side lengths using the distance formula

Using the distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ , the squared distance is $(x_2 - x_1)^2 + (y_2 - y_1)^2$ . Applying this formula gives $AB^2 = (a - 1)^2 + 1$ , $BC^2 = 1 + (1 - b)^2$ , and $AC^2 = a^2 + b^2$ .

Step 3: Substitute the squared lengths into the Pythagorean equation

We now substitute the expressions for $AC^2$ , $AB^2$ , and $BC^2$ directly into the equation $AC^2 = AB^2 + BC^2$ . Expanding the quadratic expressions yields $a^2 + b^2 = (a^2 - 2a + 1 + 1) + (1 + 1 - 2b + b^2)$ .

Step 4: Simplify to establish the relation between a and b

Subtracting $a^2 + b^2$ from both sides gives $0 = -2a - 2b + 4$ . Moving the variable terms to the left-hand side, we get $2a + 2b = 4$ , which simplifies to $a + b = 2$ upon dividing throughout by 2.

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