Pythagorean therorem: word problems

Question

To get to his parents' house, Austin would have to drive due north 12 kilometers. To get to his grandparents' house, he would have to drive due east 16 kilometers. What is the straight-line distance between the parents' and grandparents' houses?

Answer 1

You may also recognize that this is a 3:4:5 right triangle.

If the legs have a 3:4 ratio, the the hypotenuse has to be 5.

The distances for the legs are 12 and 16 miles.

That's 3 times 4 = 12 and 4 times 4 = 16, so the hypotenuse has to be 5 times 4 = 20.

You can use this shortcut if you have two of the sides and they fit part of the 3:4:5 ratio.

Two legs: 3:4 as above. The hypotenuse is 5.

Hypotenuse and one leg - 5:4 - The other leg is 3.

Hypotenuse and one leg - 5:3 - The other leg is 4.

3:4:5 right triangles show up in test questions fairly often, almost always with the two legs given with a 3:4 ratio. You can save a lot of time by doing a quick check. It only takes a couple of seconds and you can probably calculate the answer in your head. This is a quite useful item to have in what I call your "bag of tricks"!

Answer 2

Hi Ana! I'll try to give a simplified answer first.

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Simplified:

12

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16

12² + 16² = C²

144 + 256 = C²

400 = C²

√400 = C

√20*20 = C

√20² = C

20 = C = distance between houses

The distance between the houses is 20 kilometers.

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Detailed:

We need to make a triangle.

Let's let the north direction, 12, be the height.

Let's let the east direction, 16, be the base.

The straight line distance between the houses should be the hypotenuse.

We need to find the hypotenuse! The hypotenuse is the diagonal line.

12

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| \ ?

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16

Now, we use the Pythagorean theorem to find that hypotenuse.

A² + B² = C²

Let A equal the height and B equal the base.

12² + 16² = C²

Nice! Now, we have something we can solve. Simplify the exponents first because of PEMDAS.

144 + 256 = C²

Next, we need to add those two numbers.

400 = C²

Here comes the square root! We want to get rid of that exponent above 'C'? This is how we do it.

√400 = C

To simplify the square root, we need to find the two factors of 400 that are square. That means that the same number is multiplied by itself.

√20*20 = C

√20² = C

A square root and a squared number cancel each other out!

20 = C

C = the hypotenuse, so the hypotenuse is 20.

The hypotenuse is the distance between the two houses.

The distance between the two houses is 20 kilometers.