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To measure the distance across a pond, a surveyor locates points A, B, C, D, and E as shown. Find the length of AB to the nearest meter. Explain how you got this answer.

Answer:

formula with a squared term.

It’s not about distance in the sense of walking diagonally across a room. It’s about any distance, like the “distance” between our movie preferences or colors.

If it can be measured, it can be compared with the Pythagorean Theorem. Let’s see why.

Understanding The Theorem

We agree the theorem works. In any right triangle:

pythagorean theorem

If a=3 and b=4, then c=5. Easy, right?

Well, a key observation is that a and b are at right angles (notice the little red box). Movement in one direction has no impact on the other.

It’s a bit like North/South vs. East/West. Moving North does not change your East/West direction, and vice-versa — the directions are independent (the geek term is orthogonal).

The Pythagorean Theorem lets you use find the shortest path distance between orthogonal directions. So it’s not really about right triangles — it’s about comparing “things” moving at right angles.

You: If I walk 3 blocks East and 4 blocks North, how far am I from my starting point?

Me: 5 blocks, as the crow flies. Be sure to bring adequate provisions for your journey.

You: Uh, ok.

So what is “c”?

Well, we could think of c as just a number, but that keeps us in boring triangle-land. I like to think of c as a combination of a and b.

But it’s not a simple combination like addition — after all, c doesn’t equal a + b. It’s more a combination of components — the Pythagorean theorem lets us combine orthogonal components in a manner similar to addition. And there’s the magic.

In our example, C is 5 blocks of “distance”. But it’s more than that: it contains a combination of 3 blocks East and 4 blocks North. Moving along C means you go East and North at the same time. Neat way to think about it, eh?

From the picture attached,

        DE = 5 m, CD = 5 m, CE = 4 m and CA = 12 m

In ΔCDE and ΔCBA,

Two sides AB and ED are parallel and two lines AC and BC are intersecting these lines,

Therefore, ∠CDE ≅ ∠CBA and ∠CED ≅ ∠CAB [Corresponding angles]

By AA postulate of similarity,

ΔCDE ~ ΔCBA.

By using the property of two similar triangles,

   "Corresponding sides of two similar triangles are proportional"

[tex]\frac{CE}{CA}= \frac{DE}{BA}[/tex]

[tex]\frac{4}{12}= \frac{5}{AB}[/tex]

[tex]AB=\frac{5\times 12}{4}[/tex]

[tex]AB=15[/tex] m

      Therefore, length of AB will be 15 meters.

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