Question

If GF is a midsegment of CDE, find CD.

A. 3.4

B. 6.8

C. 13.6

D. 14

Answer 1

Answer:

C

Step-by-step explanation:

Triangle CGF and triangle CED are similar. Hence, the ratio of their corresponding sides are equal. Thus we can write:

$\frac{5x+4}{2x+3}=\frac{CD}{3x+0.8}$

We can now cross multiply and solve for CD:

$\frac{5x+4}{2x+3}=\frac{CD}{3x+0.8}\\(5x+4)(3x+0.8)=(2x+3)(CD)\\15x^2+4x+12x+3.2=(2x+3)(CD)\\15x^2+16x+3.2=(2x+3)(CD)\\CD=\frac{15x^2+16x+3.2}{2x+3}$

Since GF is a midsegment of CDE, CD is double of CF. So we can write:

$CD=2CF\\\frac{15x^2+16x+3.2}{2x+3}=2(3x+0.8)\\\frac{15x^2+16x+3.2}{2x+3}=6x+1.6\\15x^2+16x+3.2=(6x+1.6)(2x+3)\\15x^2+16x+3.2=12x^2+18x+3.2x+4.8\\15x^2+16x+3.2=12x^2+21.2x+4.8\\3x^2-5.2x-1.6=0$

By using quadratic formula $\frac{-b+-\sqrt{b^2-4ac} }{2a}$ and with a=3, b= -5.2, and c= -1.6, we find the value of x to be:

$\frac{5.2+-\sqrt{(-5.2)^2-4(3)(-1.6)} }{2(3)}=2$

Since the expression for CD is $\frac{15x^2+16x+3.2}{2x+3}$ , we plug in $x=2$ into this expression to find value of CD:

$\frac{15(2)^2+16(2)+3.2}{2(2)+3}=13.6$

The correct answer is C

Answer 2

Answer:

C. 13.6

Step-by-step explanation:

We have been given that GF is a mid-segment of CDE.

Since we know that mid-segment of a triangle is half the length of its parallel side.

We can see that ED is parallel to GF , so measure of GF will be half the measure of ED. We can represent this information as:

$GF=\frac{1}{2}ED$

Let us substitute given value of GF and ED to find our x.

$2x+3=\frac{1}{2}(5x+4)$

Multiply both sides of equation by 2.

$2*(2x+3)=2*\frac{1}{2}(5x+4)$

$2*(2x+3)=5x+4$

$4x+6=5x+4$

$6=5x+4-4x$

$6=x+4$

$6-4=x$

$2=x$

We can see that triangle CFG is similar to triangle CDE, so we will use proportions to find length of CD.

$\frac{CF}{GF}=\frac{CD}{ED}$

Substitute given values.

$\frac{3x+0.8}{2x+3}=\frac{CD}{5x+4}$

Upon substituting x=2 in our equation we will get,

$\frac{3*2+0.8}{2*2+3}=\frac{CD}{5*2+4}$

Let us simplify our equation.

$\frac{6+0.8}{4+3}=\frac{CD}{10+4}$

$\frac{6.8}{7}=\frac{CD}{14}$

$14*\frac{6.8}{7}=CD$

$2*6.8=CD$

$13.6=CD$

Therefore, CD equals 13.6 and option C is the correct choice.