Question

if the length of one of the legs of a right triangle is 10 and the length of the other leg is 24, what is the length of the hypotenuse?
17
13
169
26
multiple choice 2 points
which set of side measurements could be those of a triangle?
3cm, 4cm, 7cm
2cm, 3cm, 5cm
4cm, 5cm, 10cm
5cm, 7cm, 11cm
multiple choice 2 points
the sides of △abc are 2, 3, and 4. which set of numbers could represent the sides of a triangle similar to △abc?
{5,6,7}
{6,9,16}
{20,30,40}
{12,13,14}

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a = 10\) and \(b = 24\), the Pythagorean theorem is \(c^{2}=a^{2}+b^{2}\), where \(c\) is the hypotenuse. So \(c^{2}=10^{2}+24^{2}\).

Step2: Calculate squares

\(10^{2}=100\) and \(24^{2}=576\), then \(c^{2}=100 + 576=676\).

Step3: Find the square - root

\(c=\sqrt{676}=26\).

Step4: Recall triangle inequality theorem

The triangle inequality theorem states that for a triangle with side lengths \(x\), \(y\), and \(z\), \(x + y>z\), \(x+z > y\), and \(y + z>x\).
For \(3\mathrm{cm},4\mathrm{cm},7\mathrm{cm}\), \(3 + 4=7\), does not satisfy the theorem.
For \(2\mathrm{cm},3\mathrm{cm},5\mathrm{cm}\), \(2+3 = 5\), does not satisfy the theorem.
For \(4\mathrm{cm},5\mathrm{cm},10\mathrm{cm}\), \(4 + 5<10\), does not satisfy the theorem.
For \(5\mathrm{cm},7\mathrm{cm},11\mathrm{cm}\), \(5+7>11\), \(5 + 11>7\), \(7+11>5\), satisfies the theorem.

Step5: Recall similarity of triangles

For two similar triangles, the ratios of their corresponding side lengths are equal.
For \(\triangle ABC\) with side lengths \(2\), \(3\), and \(4\), in the set \(\{20,30,40\}\), \(\frac{20}{2}=\frac{30}{3}=\frac{40}{4}=10\).

Answer:

1. D. 26
2. D. \(5\mathrm{cm},7\mathrm{cm},11\mathrm{cm}\)
3. C. \(\{20,30,40\}\)