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- [Tutor] Pause this video
and see if you can find

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the area of this triangle,

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and I'll give you two hints.

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Recognize, this is an isosceles triangle,

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and another hint is that
the Pythagorean Theorem

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might be useful.

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Alright, now let's work
through this together.

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So, we might all remember
that the area of a triangle

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is equal to one half times
our base times our height.

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They give us our base.

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Our base right over here is,

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our base is 10.

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But what is our height?

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Our height would be,

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let me do this in another color,

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our height would be the length
of this line right over here.

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So, if we can figure that out,

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then we can calculate what
one half times the base 10

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times the height is.

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But how do we figure out this height?

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Well, this is where
it's useful to recognize

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that this is an isosceles triangle.

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An isosceles triangle has
two sides that are the same.

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And so, these base angles are
also going to be congruent.

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And so, and if we drop an
altitude right over here

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which is the whole
point, that's the height,

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we know that this is, these
are going to be right angles.

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And so, if we have two triangles

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where two of the angles are the same,

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we know that the third angle
is going to be the same.

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So, that is going to be congruent to that.

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And so, if you have two triangles,

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and this might be obvious
already to you intuitively,

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where look, I have two angles in common

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and the side in between them is common,

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it's the same length,

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well that means that these are going to be

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congruent triangles.

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Now, what's useful about
that is if we recognize

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that these are congruent triangles,

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notice that they both have a side 13,

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they both have a side, whatever
this length in blue is.

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And then, they're both
going to have a side length

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that's half of this 10.

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So, this is going to be five,
and this is going to be five.

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How was I able to deduce that?

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You might just say, oh that
feels intuitively right.

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I was a little bit more rigorous here,

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where I said these are
two congruent triangles,

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then we're going to split this 10 in half

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because this is going to be equal to that

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and they add up to 10.

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Alright, now we can use
the Pythagorean Theorem

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to figure out the length of
this blue side or the height.

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If we call this h, the
Pythagorean Theorem tells us

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that h squared plus five
squared is equal to 13 squared.

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H squared plus five squared,

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plus five squared is going
to be equal to 13 squared,

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is going to be equal to our longest side,

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our hypotenuse squared.

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And so, let's see.

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Five squared is 25.

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13 squared is 169.

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We can subtract 25 from both sides

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to isolate the h squared.

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So, let's do that.

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And what are we left with?

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We are left with h squared is equal to

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these canceled out, 169 minus 25 is 144.

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Now, if you're doing it
purely mathematically,

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you say, oh h could be plus or minus 12,

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but we're dealing with the distance,

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so we'll focus on the positive.

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So, h is going to be equal
to the principal root of 144.

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So, h is equal to 12.

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Now, we aren't done.

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Remember, they don't want us to just

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figure out the height here,

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they want us to figure out the area.

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Area is one half base times height.

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Well, we already figured out

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that our base is this 10 right over here,

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let me do this in another color.

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So, our base is that distance which is 10,

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and now we know our height.

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Our height is 12.

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So, now we just have to compute
one half times 10 times 12.

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Well, that's just going to be equal to

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one half times 10 is five,

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times 12 is 60,

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60 square units, whatever
our units happen to be.

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That is our area.