Answer:
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The height is: "4 inches" ;
and the base length is: "12 inches" .
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Step-by-step explanation:
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The formula of the area of a triangle is:
Area = (1/2) * (base length) * (perpendicular height) ; or write as:
A = (1/2) * (b) * (h) ;
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Given: A = 24 in² ;
b = h + 8 ;
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Find: "b" ; and: "h" ;
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→ Since: " A = (1/2) * b * h " ;
→ Plug in our known values:
→ 24 = (1/2) * (h + 8) * h ;
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Find: h ;
Find "b" , which is: "(h +8)" ;
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We have:
→ 24 = (1/2) * (h + 8) * h ;
Multiply EACH SIDE of the equation by "2" ; to get rid of the fraction:
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→ 2 * {24 = (1/2) * (h + 8) * h} ;
→ to get:
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→ 48 = 1 * (h + 8) * h ;
↔ Rewrite: h(h + 8) = 48 ;
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Note the "distributive property of multiplication":
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a(b+c) = ab + ac ;
a(b−c) = ab − ac ;
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→ So; h(h + 8) = h*h + h*8 = h² +8h = 48 ;
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We have: " h² + 8h = 48 " ; To solve for "h" ; let us see if we can
write this equation in "quadratic format" ; that is:
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" ax² + bx + c = 0 ; a ≠ 0 ; " ;
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We have: h² + 8h = 48 ; Subtract "48" from EACH SIDE of the equation:
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h² + 8h − 48 = 48 − 48 ;
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to get: h² + 8h − 48 = 0 ;
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Note that is equation IS, in fact, written in "quadratic format" ;
that is: "ax² + bx + c = 0 ; a ≠ 0 " ;
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in which: a = 1 ;
(Note: The "implied coefficient" of "1"; since anything multipled by "1" is that same result);
b = 8 ;
c = - 48;
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Now, let us see if we can solve by factoring; if we cannot, we can use the quadratic equation formula:
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Let us trying factoring: h² + 8h − 48 = (h+12) (h − 4) = 0 ;
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Since anything multiplied by "zero" equals "zero" ;
Then either: (h+12) = 0 ; h = -12 ;
(h − 4) = 0 ; h = 4 ;
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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 "
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So, h = 4 ; b = 12.
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Now check our work: "A = (1/2) (b) (h)" ; Given "A = 24" .
24 = (1/2) (12) (4)? 24 = (1/2) * 48 ? YES!
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So, the height is: "4 inches" ;
and the base length is: "12 inches" .
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