Find the value of x in the figure – can you solve SIMILAR TRIANGLE problems?

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How to solve similar triangle problems using the AA Theorem.

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The upper triangle is similar to the whole triangle, but smaller by factor 2, thus X=5. The answer would have been more obvious if the drawing had not been misleading (the upper 6 is longer than the lower 6).

chrisw

Similar triangles: 12 is to 10 as 6 is to x. 12/10=6/x. Reduce: 6/5=6/x. Substitute: 6/5=6/5. x=5.

toughenupfluffy

I definitely did this the hard way. I observed that line segment X would bisect the hypotenuse of the big triangle, so calculated the length of the hypotenuse as 2 root 61, so length of hypotenuse in smaller triangle was root 61, subtracting 6^2 (36) from 61 was 25, and root 25 was 5, length of X.

davidloewen

Many took a stab at it, noticed it was at the halfway point vertically on a right triangle, which he considered as half a rectangle that was 10 units wide x 12 units high, and Many came up with the answer 5. He's a clever guy!

SeekingHisWill

I used Pythagoras' theorem..

Outer triangle:
12 square + 10 square = hypotenuse square
hypotenuse square = 244
hypotenuse = 15.62

Inner triangle:
(Inner hypotenuse = 0.5 * Outer hypotenuse)
6 square + x square = (0.5 * 15.62) square
x square = 7.81 square - 6 square
x square = 60.9961 - 36.00 = 24.9961
x = 4.9996 (approx 5)

chaitanyakonher

It was an interesting looking puzzle but I figured since the larger triangle is a right triangle it's half of a rectangle with base 10. If line X extended all the way to the other side of the rectangle it would also be 10 in length. If you draw a line from the bottom corner to the opposite top corner of that rectangle it will pass through the extension of line X at the halfway point. Half of 10 is 5

niminitzar

Greetings. The answer is 10/12=X/6
using the properties of similar triangles. That is 60=12X, and X=5
after dividing both sides by 12.
The answer is 5 units.

devonwilson

Solution:

If two triangles have the same angles - as they do in this case - the ratio between their sides is identical.
Or written more mathematical:
If α₁ = α₂ and β₁ = β₂ and γ₁ = γ₂ then
a₁/b₁ = a₂/b₂ and a₁/c₁ = a₂/c₂ and b₁/c₁ = b₂/c₂
and obviously also their inverses
b₁/a₁ = b₂/a₂ and c₁/a₁ = c₂/a₂ and c₁/b₁ = c₂/b₂

You can also write it as a direct comparison between the sides:
a₁/a₂ = b₁/b₂ = c₁/c₂

As such, we can say, that:
10 / (6 + 6) = x / 6 |*6
5 * 2 * 6 / (2 * 6) = x
x = 5

m.h.

The use of the term cross product does not apply in this context (14:31). The term applies to vector multiplication, and finding the perpendicular vector. You mean "cross multiply" which sounds like cross product. There is also no such thing as cross cancel. When the 2 fractions can be reduced by dividing a common factor into both numerator and denominator, I believe that is called "canceling", not cross cancel. So to solve this simple proportion, you can cancel by dividing the 6 and the 10 by 2, which is common to both. Thank you for your wonderful vids!

xlwatcher

Is it true that the hypotenuse side of the small triangle will be 1/2 the size of the big triangle?

michaelmappin

The badly drawn diagram threw me at first but the I realised that x must be 5 as the right hand side is 12 and the proportions are the same for a similar triangle such that 12 becomes 6 and 10 becomes 5.
Notice how the bottom left point of the triangle is not extended properly to a point and the right-hand side divides slightly lower than half way up. Very tricky drawing to put us off.

TheSimCaptain

5, because it’s in the half of the triangle. Concluding, no squares nor roots. Just looking at the pic.

cromssie

That is basis triangle theory as the top angle is the same and so the tangent is the same. 10/12=x/6. So x=5.

danieldennis

Jeez, my brain's like you're knocking on your buddies' front door to hang out and instead he makes you hop the side fence, cross the alligator moat and solve the troll's riddle before you can begin lockpicking the back door, both brain and beer completely melted by the time you get inside 😅

Anyone else get knee deep in ridiculous never-ending decimals thinking you're waaaay off track, the troll key-in-hand snickering at you while you overzealously finesse this bathroom stall lock? Then as soon as 5 bursts in late, you notice it was just a simple cross-division problem?

Can easily see at least 3 different ways to get in this dude's house that would've been so much faster; open window, doggy door, freakin sledgehammer to the living room wall, ya know what screw it, I'm goin home

MindfulPond