1. Remember that the converse of the Pythagorean theorem says that if the square of the longest side of a triangle is equal to the sum of the squares of its smaller side, the triangle is a right one:
We know from our problem that the longest side is 106, and the smaller sides are 56 and 90, so:
[tex]106^2=56^2+90^2[/tex]
[tex]11236=3136+8100[/tex]
[tex]11236=12236[/tex]
We can conclude that our triangle is a right triangle. The correct answer is the first choice: right
2. First, we are going to find [tex]y[/tex] using the fact that the two triangles are similar:
[tex] \frac{7}{y} = \frac{y}{4} [/tex]
[tex]y^2=28[/tex]
[tex]y= \sqrt{28} [/tex]
Next, we are going to use the Pythagorean theorem to find [tex]x[/tex]:
[tex]x^2= 4^2+( \sqrt{28} )^2[/tex]
[tex]x^2=16+28[/tex]
[tex]x^2=44[/tex]
[tex]x= \sqrt{44} [/tex]
[tex]x=6.63[/tex]
We can conclude that the correct answer is the third choice: 6.63
3. To solve this, we are going to use the Pythagorean theorem one more time. We know that the longest side of our triangle is 25, and the shorter sides are 7 and [tex]x[/tex], so:
[tex]25^2=7^2+x^2[/tex]
[tex]625=49+x^2[/tex]
[tex]x^2=576[/tex]
[tex]x= \sqrt{576} [/tex]
[tex]x=24[/tex]
We can conclude that the correct answer is the first choice: 24
