# Question 1b82b

Mar 29, 2017

Tina would need a minimum of $54$in of wood for DFG, dependent on its thickness. Thicker frames would need longer lengths to accommodate the three corners that project beyond the art pieces.

#### Explanation:

The two triangles are $s i m i l a r$ because they both share a common angle, and the given sides of one are similar multiples of the given sides of the other.

Then, by the properties of similar triangles:

Since side $D F = 3 \cdot A B , \mathmr{and} B C = 3 \cdot F G , t h e n D G = 3 \cdot A C$

So $D G = 3 \cdot 7$in $= 21$in.

Perimeter of DFG = 15+18+21 = 54in.
This is the correct answer and a correct method for the solution.

There is also a method of solving the larger triangle with a formula to find the value of the common angle using the three given sides of the smaller triangle:

$\cos B = \frac{{c}^{2} + {a}^{2} - {b}^{2}}{2 c a} = \frac{{5}^{2} + {6}^{2} - {7}^{2}}{2 \times 6 \cdot 5} = 0.2$

By the way that angle is $78.5 \mathrm{de} g r e e s$.

Then we can apply the same formula to the larger triangle:

$\cos B = \frac{{c}^{2} + {a}^{2} - {b}^{2}}{2 c a} = \frac{{15}^{2} + {18}^{2} - {b}^{2}}{2 \times 18 \cdot 15} = 0.2$

$\frac{{15}^{2} + {18}^{2} - {b}^{2}}{2 \times 18 \cdot 15} = 0.2$

$b = 21 = D G$

Mar 29, 2017

$131.76$ ${\text{in}}^{2}$

#### Explanation:

When we see the triangles, we can say that the triangles are similar to each other. The corresponding sides are in multiples of their common sides.

We can say that,

color(orange)("AB"xx3="DF"

color(orange)("BC"xx3="FG"

So,

color(orange)("DG"=7xx3=21

Now we know the three sides of the triangle.We can find the area of the triangle using the Heron's formula

color(blue)("Area"=sqrt(s(s-a)(s-b)(s-c))

Where, $a , b \mathmr{and} c$ are the sides and $s = \frac{a + b + c}{2}$

color(blue)(s=(15+18+21)/2=27

Let's apply the values to the formula

$\rightarrow \sqrt{27 \left(27 - 15\right) \left(27 - 18\right) \left(27 - 21\right)}$

$\rightarrow \sqrt{27 \left(12\right) \left(9\right) \left(6\right)}$

Split into prime factors

$\rightarrow \sqrt{\underbrace{3 \cdot 3} \cdot 3 \cdot \underbrace{2 \cdot 2} \cdot \underbrace{3 \cdot 3} \cdot \underbrace{3 \cdot 3} \cdot 2}$

$\rightarrow 3 \cdot 3 \cdot 3 \cdot 2 \sqrt{3 \cdot 2}$

color(green)(rArr54sqrt6

color(green)(~~131.76

So, the minimum area of wood required is $131.76$ ${\text{in}}^{2}$

Hope this helps...! :)