Question

On paper, sketch 2 sizes of similar right triangles, each having one of its non-right angles as a 45 degree angle. On the smaller triangle, label the hypotenuse with a length of 102–√ and the side opposite the 45 degree angle with a length of 10. On the larger triangle, label the side opposite the 45 degree angle with a length of 14. Now, determine the length of the hypotenuse of the larger triangle. (Can write answer in radical form OR rounded to the nearest tenth. I will check this answer for every student and I will give accurate points and feedback.) (1 point) The hypotenuse of the larger triangle has a length of

Answer 1

In a 45-45-90 triangle, the relationship between the lengths of the sides is that the lengths of the legs are equal, and the length of the hypotenuse is equal to the length of a leg multiplied by \(\sqrt{2}\).

For the smaller triangle:
- You have one leg (the side opposite the 45-degree angle) with a length of 10.
- The length of the hypotenuse is given as \(10\sqrt{2}\), since the hypotenuse is \(\text{leg} \times \sqrt{2} = 10\sqrt{2}\).

Now for the larger triangle:
- The side opposite the 45-degree angle has a length of 14.
- Using the same property for 45-45-90 triangles, the length of the hypotenuse will be \(14\sqrt{2}\).

Now, if you'd like to provide the hypotenuse in both radical form and in decimal form:
- The hypotenuse is \(14\sqrt{2}\).
- To find the decimal approximation, calculate \(14\cdot\sqrt{2}\). The approximate value of \(\sqrt{2}\) is about 1.414.

Calculating:
\[
14 \times 1.414 \approx 19.396
\]

Rounded to the nearest tenth:
\[
\approx 19.4
\]

Thus, the length of the hypotenuse of the larger triangle is:
- In radical form: \(14\sqrt{2}\).
- Rounded to the nearest tenth: \(19.4\).

You can choose either format based on your preference.