Answer: _______________________________________________ The height is: "4 inches" ; and the base length is: "12 inches" . ____________________________________ Explanation: ____________________________________ The formula of the area of a triangle is:
Area = (1/2) * (base length) * (perpendicular height) ; or write as:
A = (1/2) * (b) * (h) ; _________________________________ Given: A = 24 in² ; b = h + 8 ; ________________________ Find: "b" ; and: "h" ; _____________________________ → Since: " A = (1/2) * b * h " ;
→ Plug in our known values:
→ 24 = (1/2) * (h + 8) * h ; _________________________________ Find: h ; Find "b" , which is: "(h +8)" ; _________________________________ We have:
→ 24 = (1/2) * (h + 8) * h ;
Multiply EACH SIDE of the equation by "2" ; to get rid of the fraction: _______________________________________________________ → 2 * {24 = (1/2) * (h + 8) * h} ;
→ to get: ______________________________ → 48 = 1 * (h + 8) * h ;
↔ Rewrite: h(h + 8) = 48 ; _________________________________________ Note the "distributive property of multiplication": _________________________________________ a(b+c) = ab + ac ; a(b−c) = ab − ac ; _________________________________________ → So; h(h + 8) = h*h + h*8 = h² +8h = 48 ; ______________________________________ We have: " h² + 8h = 48 " ; To solve for "h" ; let us see if we can write this equation in "quadratic format" ; that is: _______________________________________________ " ax² + bx + c = 0 ; a ≠ 0 ; " ; _________________________________________ We have: h² + 8h = 48 ; Subtract "48" from EACH SIDE of the equation: _________________________________________ h² + 8h − 48 = 48 − 48 ; _________________________________________ to get: h² + 8h − 48 = 0 ; _________________________________________ Note that is equation IS, in fact, written in "quadratic format" ; that is: "ax² + bx + c = 0 ; a ≠ 0 " ; _________________________________________ in which: a = 1 ; (Note: The "implied coefficient" of "1"; since anything multipled by "1" is that same result); b = 8 ; c = - 48; _____________________________________ Now, let us see if we can solve by factoring; if we cannot, we can use the quadratic equation formula: _____________________________________ Let us trying factoring: h² + 8h − 48 = (h+12) (h − 4) = 0 ; ________________________________________________ Since anything multiplied by "zero" equals "zero" ;
Then either: (h+12) = 0 ; h = -12 ; (h − 4) = 0 ; h = 4 ; ________________________________________________ So we have two (2) values for "h" ; "h = 4" , and "h = -12" .
So, which value do we use for "h"? Since "h" refer to "height"; we know that "height" cannot be a "negative value"; so we use:
"h = 4" .
Now, we are given: "b = h + 8 = 4 + 8 = 12 " _______________________________________ So, h = 4 ; b = 12. ______________________ Now check our work: "A = (1/2) (b) (h)" ; Given "A = 24" .
24 = (1/2) (12) (4)? 24 = (1/2) * 48 ? YES! ______________________________________ So, the height is: "4 inches" ; and the base length is: "12 inches" . ______________________________________
For graphing purposes, it is often convenient to rewrite the equation so the solutions are where the function is zero. Here, we can do that by subtracting 13 from the equation.
