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" . ______________________________________
