The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram. Find the value of $x$. (ii) Verify, with this value of $x$, that the lengths of the triangle above form a pythagorean triple.

Leaving Cert Mathematics Algebra: The lengths of the sides of a ri

The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram - Leaving Cert Mathematics - Question 5 - 2016

Question 5

The-lengths-of-the-sides-of-a-right-angled-triangle-are-given-by-the-expressions-$x---1$,-$4x$,-and-$5x---9$,-as-shown-in-the-diagram-Leaving Cert Mathematics-Question 5-2016.png

The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram. Find the value of $x$. (ii) V... show full transcript

Worked Solution & Example Answer:The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram - Leaving Cert Mathematics - Question 5 - 2016

Step 1

Find the value of $x$

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Answer

To find the value of $x$ , we set up the Pythagorean theorem:

$(5x - 9)^2 = (x - 1)^2 + (4x)^2$

Expanding each term:

Left-hand side: $(5x - 9)^2 = 25x^2 - 90x + 81$
Right-hand side:
$(x - 1)^2 = x^2 - 2x + 1$ $(4x)^2 = 16x^2$

Combining the right-hand side: $x^2 - 2x + 1 + 16x^2 = 17x^2 - 2x + 1$

Setting both sides equal: $25x^2 - 90x + 81 = 17x^2 - 2x + 1$ Now we bring all terms to one side: $25x^2 - 17x^2 - 90x + 2x + 81 - 1 = 0$ This simplifies to: $8x^2 - 88x + 80 = 0$ Dividing by 8: $x^2 - 11x + 10 = 0$ Factoring gives: $(x - 1)(x - 10) = 0$ Thus, we have two possible solutions: $x = 1 \text{ or } x = 10$ Given the expressions for side lengths, the valid solution is $x = 10$ .

Step 2

Verify, with this value of $x$, that the lengths of the triangle above form a pythagorean triple.

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Answer

Substituting $x = 10$ into the expressions for the sides:

Side 1: $x - 1 = 10 - 1 = 9$
Side 2: $4x = 4(10) = 40$
Side 3: $5x - 9 = 5(10) - 9 = 41$

Now, we check if these lengths satisfy the Pythagorean theorem:

$9^2 + 40^2 = 41^2$ Calculating each term:

$9^2 = 81$
$40^2 = 1600$
$41^2 = 1681$

Now we sum the squares of the two shorter sides: $81 + 1600 = 1681$ Since both sides equal, we confirm: $9^2 + 40^2 = 41^2$

Thus, the sides 9, 40, and 41 do indeed form a Pythagorean triple.

The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram - Leaving Cert Mathematics - Question 5 - 2016

Worked Solution & Example Answer:The lengths of the sides of a right-angled triangle are given by the expressions $x - 1$, $4x$, and $5x - 9$, as shown in the diagram - Leaving Cert Mathematics - Question 5 - 2016

Find the value of $x$

Verify, with this value of $x$, that the lengths of the triangle above form a pythagorean triple.

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