Question

15 do side lengths of 7, 8, and 10 create a right triangle? show your work to prove it.
answer
16 the bottom of a ladder must be placed 4 feet from a wall. the ladder is 15 feet long. how far above the ground does the ladder touch the wall? (hint: draw a picture first!)
height:
17 a soccer field is a rectangle 110 meters wide and 140 meters long. the coach asks players to run from one corner to the other corner diagonally across. what is that distance? (hint: draw a picture first!)
distance

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle, \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side). Let \(a = 7\), \(b = 8\), and \(c = 10\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\).

$$a^{2}+b^{2}=7^{2}+8^{2}=49 + 64=113$$

$$c^{2}=10^{2}=100$$

Since \(a^{2}+b^{2}
eq c^{2}\), the side - lengths of 7, 8, and 10 do not create a right - triangle.

Step2: For the ladder problem

The ladder, the ground, and the wall form a right - triangle. The distance from the wall to the bottom of the ladder is \(a = 4\) feet and the length of the ladder is \(c = 15\) feet. Let the height on the wall be \(b\). Using the Pythagorean theorem \(b=\sqrt{c^{2}-a^{2}}\).

$$b=\sqrt{15^{2}-4^{2}}=\sqrt{225 - 16}=\sqrt{209}\approx14.46$$

feet

Step3: For the soccer - field problem

The length and width of the soccer field form the two legs of a right - triangle, and the diagonal is the hypotenuse. Let \(a = 110\) meters and \(b = 140\) meters. Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\).

$$c=\sqrt{110^{2}+140^{2}}=\sqrt{12100+19600}=\sqrt{31700}\approx178.05$$

meters

Answer:

1. No
2. \(\sqrt{209}\approx14.46\) feet
3. \(\sqrt{31700}\approx178.05\) meters