The problem provides an equation relating the sides a, b, and c of a triangle ABC: $(a - 8)^2 + (b - 15)^2 + (c - 17)^2 = 0$. We need to determine the type of triangle based on this information.

Analyzing the Equation for Triangle Sides

The given equation is a sum of squared terms equal to zero. In real numbers, the square of any real number is non-negative ($(x)^2 \ge 0$). The only way for the sum of non-negative terms to be zero is if each individual term is zero.

Therefore, we must have:

• $(a - 8)^2 = 0$
• $(b - 15)^2 = 0$
• $(c - 17)^2 = 0$

Solving for Sides a, b, and c

Solving each equation for the respective side length:

• From $(a - 8)^2 = 0$, taking the square root of both sides gives $a - 8 = 0$, so $a = 8$.
• From $(b - 15)^2 = 0$, taking the square root of both sides gives $b - 15 = 0$, so $b = 15$.
• From $(c - 17)^2 = 0$, taking the square root of both sides gives $c - 17 = 0$, so $c = 17$.

So, the lengths of the sides of triangle ABC are $a=8$, $b=15$, and $c=17$.

Checking Triangle Inequality

Before determining the type of triangle, we should verify that these side lengths can actually form a triangle. The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

• $a + b > c$: $8 + 15 = 23$. Is $23 > 17$? Yes.
• $a + c > b$: $8 + 17 = 25$. Is $25 > 15$? Yes.
• $b + c > a$: $15 + 17 = 32$. Is $32 > 8$? Yes.

Since all three inequalities hold, sides with lengths 8, 15, and 17 can form a valid triangle.

Determining Triangle Type

Now we classify the triangle based on its side lengths (8, 15, 17).

• Equilateral triangle: All sides are equal ($a=b=c$). Here, $8 \ne 15 \ne 17$, so it's not equilateral.
• Isosceles triangle: At least two sides are equal. Here, no two sides are equal ($8 \ne 15$, $8 \ne 17$, $15 \ne 17$), so it's not isosceles.
• Right-angled triangle: The sides satisfy the Pythagorean theorem ($a^2 + b^2 = c^2$, or permutations, where the longest side is the hypotenuse). The longest side is 17. Let's check if $8^2 + 15^2 = 17^2$.

  Calculate the squares:

  • $8^2 = 64$
  • $15^2 = 225$
  • $17^2 = 289$

  Check the sum of the squares of the two shorter sides:

  \($8^2 + 15^2 = 64 + 225 = 289\)

  Compare this to the square of the longest side:

  \($17^2 = 289\)

  Since \(8^2 + 15^2 = 17^2\) (\(289 = 289\)), the Pythagorean theorem holds.

• Obtuse angled triangle: The square of the longest side is greater than the sum of the squares of the other two sides ($c^2 > a^2 + b^2$). Since $17^2 = 8^2 + 15^2$, it is not an obtuse triangle.

Because the sides satisfy the Pythagorean theorem, the triangle ABC is a right-angled triangle.