Answer:
[tex]\textsf{Statement 2.}\quad \boxed{ce=a^2\; and \;cf=b^2}[/tex]
[tex]\textsf{Reason 3.}\quad\boxed{\textsf{Addition Property of Equality}}[/tex]
[tex]\textsf{Reason 4.}\quad \boxed{\textsf{Substitution Property of Equality}}[/tex]
[tex]\textsf{Statement 5.}\quad\boxed{c(e+f)=a^2+b^2}[/tex]
[tex]\textsf{Reason 5.}\quad \boxed{\textsf{Distributive Property}}[/tex]
[tex]\textsf{Statement 7.}\quad \boxed{c\cdot c=a^2+b^2}[/tex]
[tex]\textsf{Reason 8.}\quad \boxed{\textsf{Simplify}}[/tex]
Step-by-step explanation:
Statement 2
The Geometric Means Theorem (Leg Rule) states that if an altitude is drawn from the right angle of a right triangle to its hypotenuse, it divides the hypotenuse into two segments. The ratio of the length of the hypotenuse to one of the legs is equal to the ratio of that leg to the segment of the hypotenuse adjacent to it.
[tex]\boxed{\sf \dfrac{Hypotenuse}{Leg\:1}=\dfrac{Leg\:1}{Segment\;1}}\quad \sf and \quad \boxed{\sf \dfrac{Hypotenuse}{Leg\:2}=\dfrac{Leg\:2}{Segment\;2}}[/tex]
In the given triangle ABC, the altitude CD divides the hypotenuse c into two segments labelled e and f. The leg adjacent to segment e is leg a, and the leg adjacent to segment f is leg b. Therefore, according to the Geometric Means Theorem:
[tex]\dfrac{c}{a}=\dfrac{a}{e} \implies ce=a^2[/tex]
[tex]\dfrac{c}{b}=\dfrac{b}{f} \implies cf=b^2[/tex]
Therefore, statement 2 is:
[tex]\Large\boxed{ce=a^2\; and \;cf=b^2}[/tex]
[tex]\dotfill[/tex]
Reason 3
The Addition Property of Equality states that if the same quantity is added to both sides of an equation, the equality between the two sides remains unchanged.
Therefore, if we add b² to both sides of the equation ce = a² from Statement 2, we get:
[tex]ce+b^2=a^2+b^2[/tex]
Therefore, reason 3 is:
[tex]\Large\boxed{\textsf{Addition Property of Equality}}[/tex]
[tex]\dotfill[/tex]
Reason 4
The Substitution Property of Equality states that if two quantities are equal, one can be replaced by the other in any expression or equation without altering the value or validity of the expression or equation.
If we substitute the equation cf = b² (from Statement 2) into the left side of equation ce + b² = a² + b² (from Statement 3), we get:
[tex]ce + cf = a^2 + b^2[/tex]
Therefore, reason 4 is:
[tex]\Large\boxed{\textsf{Substitution Property of Equality}}[/tex]
[tex]\dotfill[/tex]
Statement 5 and Reason 5
The Distributive Property states that multiplying a number by a sum (or difference) is the same as multiplying the number by each term of the sum (or difference) and then adding (or subtracting) the results.
So, we can apply the Distributive Property by factoring out the common factor 'c' from the left side of the equation ce + cf = a² + b² (from Statement 4):
[tex]c(e+f)=a^2+b^2[/tex]
Therefore, statement 5 is:
[tex]\Large\boxed{c(e+f)=a^2+b^2}[/tex]
So, reason 5 is:
[tex]\Large\boxed{\textsf{Distributive Property}}[/tex]
[tex]\dotfill[/tex]
Statement 7
The Segment Addition Postulate states that if three points B, D and A are collinear, then the length of segment BD plus the length of segment DA equals the length of segment BA.
Therefore, in this case, as BD = e, DA = f and BA = c, then:
[tex]e + f = c[/tex]
Using the Substitution Property of Equality again, we can substitute e + f = c into the left side of the equation c(e + f) = a² + b² (from Statement 5):
[tex]c(e+f)=a^2+b^2\\\\c(c)=a^2+b^2\\\\c\cdot c=a^2+b^2[/tex]
Therefore, statement 7 is:
[tex]\Large\boxed{c\cdot c=a^2+b^2}[/tex]
[tex]\dotfill[/tex]
Reason 8
Finally, we can simplify the equation from Statement 7 by expressing the product of the same quantity as its square:
[tex]c\cdot c=a^2+b^2\\\\c^2=a^2+b^2[/tex]
Therefore, reason 8 is:
[tex]\Large\boxed{\textsf{Simplify}}[/tex]
