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HL - Problem 3
The hypotenuse-leg (HL) shortcut to determine if two triangles are congruent can be used when both triangles have right angles. The hypotenuse of a right triangle is the side of the triangle opposite the right angle. If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, the two triangles are congruent.
When the sides of a triangle are given by expressions with variables it is possible to solve for the values of the variables so that the triangles are congruent. Since the hypotenuse of both triangles must be equal, set their expressions equal to one another and solve for the variable. The same can be done with the leg of the triangles to find the value of the variables that make the triangles congruent.
Geometry teachers love to bring in algebra into their class. Just because you are done with the class doesn’t mean you can’t forget it. In this problem it's saying what the values of x and y make this two triangles congruent?
Well we have a right angle which means we have right triangles and if our hypotenuse, if y equals 5x and if our two legs 2x plus 4y and 44, if those are congruent, then yes we can say that they must be congruent. So I’m going to set these two equal to each other and I’m going to set my two hypotenuses equal to each other.
So I’m going to start off by saying y must equal 5x and the other equation I’m going to write 2x plus 4y must equal 44. So here we have two variables, two equations. I’m going to solve by substitution. So I’m going to substitute in 5x for y.
So rewriting my bottom equation replacing y with 5x, now we have one equation one variable which I can solve. 2x plus 4 times 5 is 20, so I’m going to say that’s 22x equals 44 and then the last step to solve this is to divide this is to divide by 22, so you get x equals 2.
The question is asking for x and y so I’m going to substitute in to this equation. We said y equals 5x. If x is 2, we’re going to have 5 times 2 so y must equal 10. So to make this kind of jumbled mess clear to your teacher you are going to say x equals 2 and y equals 10.
So the key here was remembering that in right triangles if the hypotenuses are congruent and if a pair of legs are congruent, then the whole triangles must be congruent.