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This video is provided as supplementary material
for courses taught at Howard Community College and in this video
I'm go to do three word problems that are like the ones you typically run
into
when you're studying similar triangles. Here's the first kind of
problem.
This one usually involves the height of an
object, something like a tree, or in this case a flagpole.
So let's say we've got a flagpole, and
it's standing straight up. It's forming a right angle with the ground.
It's a sunny day and
the sun is shining on flagpole and it's casting a shadow which is 33 feet long.
Nearby is a many, and he's standing straight up.
He's six feet tall and
he's got a shadow that's four feet long.
Now,
the first I wanna show is that we really have two similar triangles.
The sun is shining down at the same angle
on both the flagpole and the man.
So we can draw a kind of picture of the angle that the sun is forming with
the flagpole and the man by drawing a dotted line from the top up the flagpole
to what would be the top of the shadow.
And another line from the man's head
to the shadow of the man's head So now I've got
two similar triangles. They're both right triangles,
So that's one pair of congruent angles, and
they both have the same angle at the top
with the sun shine down on them. So I've got two pairs of
congruent angles, which means I have two similar triangles.
I can now find the height of the flagpole.
To do that, I'm going to use the fact that for similar triangles,
the sides
are proportional. So the height of the flag pole is one side,
and I'll compare that, I'll make a ratio between
that side, H, and the height of the man, which is six,
and that should be equal to
the ratio of the lengths of the flagpole's shadow,
which is 30, over the length of the man's shadow,
which is 4. So I've got H over 6
equals 30 over 4. I'll multiply both sides of this equation by 6,
and I'll get H equals... 6 times 30 is 180
over 4. Dividing 180
by 4, I find out that the height of the flagpole
is 45 feet.
So once again, I showed that these two triangles
were similar by showing that they had
angle-angle similarity. Once I've got the fact that they're similar,
I can find out what
the length of a missing side is by using the fact that the sides
are proportional.
Here's another kind of problem. In this kind of problem we've got a lake
and you're trying to find out the length
from one end to the other without actually getting into the water.
So I'm gonna call one end of the lake A
and the other end B, and here's what we'll do.
Starting at point A, you walk 100 feet
to some point, which we'll call point P. And then you continue walking
50 feet more.
to a point that we'll call C.
You go to the other end of the lake,
point B, walk 70 feet
until you get to point P, and then walk
35 feet more to point D. You measure the distance from C to D,
and that's 40 feet.
Now if we can show that we have similar triangles, we can use these measurements
to find out
the length of the lake from A to B. So let's show that they're similar triangles.
I wanna show that triangle APB
is similar to triangle CPD.
So I want to see if I have two pairs of side
that are proportional. Well, I can take side
AP, which is 100 feet,
and put that over side CP, which is 50 feet,
that's a pair of corresponding sides, and I'll make another fraction
with side BP, 70 feet,
over side DP,
its corresponding side, which is 35 feet.
I'll set those fractions equal to each other and I'll make sure they're equal.
100 over 50 is the same as 2 over 1,
and 70 over 35 is the same as 2 over 1,
so the proportions are the same,
which means I've got two pairs of corresponding sides that are proportional.
Now there's
and angle that those sides form that's angle P,
in both cases. Angle P
is actually a pair of vertical angles. We've got a pair of vertical angles there.
So what we've got is two pairs of proportional side
and a pair of
angles, the vertical angles, that are congruent.
That means I've got two similar triangles.
Let's use that fact to figure out what the distance is from
A to B. So let's say the distance from
A to B is L, for lake.
The corresponding distance
for L would 40, that's the corresponding
side of the other triangle. And we want to set
that ratio. L over 40, equal to the ratio
for the other sides, which is just 2 over 1.
So L over 40 equals 2 over 1. If I multiply both sides of this equation
by 40, I'll get L equals 80.
So the length of the lake is 80 feet.
Once again: two similar triangles.
I proved they were similar by using side-angle-side similarity,
and then I was able to find the unknown side.
Okay, here's the third kind of problem. This usually involves a canyon or a river.
You want to find the distance from one side of the river to the other.
So let's say you're standing at point A on one side of the river
and you notice that directly across from you there's a tree. We'll that point B.
So to do this, to find the distance across the river,
you turn 90 degrees and walk along the length of the river 15 feet.
After 15 feet you put a marker down, which we'll call point P,
and then you continue walking 10 feet more. Now you turn 90 degrees again,
away from the river, walk 8 feet...
I'm sorry, I should have said that you go to point C after 10 feet.
You walk 8 feet at that 90 degree angle to point D.
And now you notice that there's a direct line
from point D, through point P,
over to point B, where the tree is. So we can show that we formed two similar
triangles.
We've got two right angles -- that's angles A and C.
We've got a pair vertical angles at P.
So what that means is with that we've got angle-angle similarity.
Once we know we've got angle-angle similarity, we can find
the distance across the river,
which I'll call r, for river.
So this side, distance r,
corresponds to distance CD,
or line segment CD, and that's 8 feet. That's the corresponding side
of the other triangle. We'll take another
pair of corresponding sides, we'll take side
AP, which is 15 feet,
and put that over the distance for side CP,
which is 10 feet. So now I've got r over 8
equals 15 over 10. I can multiply both sides of this equation by 8.
I'm going to have r equals... 8 times 15 is 120,
divided by 10.
120 divided by 10 is 12.
So the distance from point A, one side of the river, to point B,
the other side of the river, is 12 feet. And again,
we've done this by finding a pair of similar triangles.
We found they were similar by angle-angle similarity,
and then we used that similarity to find the unknown side.
Okay, I hope that helps. Take care, I'll see you next time.