In the following figure, triangle ABC and triangle DEF are similar. What is the value of x ?
Triangle ABC and triangle DEF are shown. Side AB measures 10 millimeters. Side BC measures 4 millimeters. Side CA measures 8.5 millimeters. Side DE is not labeled. Side EF measures 12 millimeters. Side FD measures x.
Well, well, well, it looks like we have a little game of "Find x" going on here! Let's see if we can crack this puzzle.
Since triangle ABC and triangle DEF are similar, we know that their corresponding sides are proportional. In other words, if we take the ratio of the lengths of the corresponding sides, it should be the same for all sides.
Let's start by comparing side AB to side DE. We have AB = 10 millimeters and DE is not labeled. Therefore, we have 10/DE.
Next, let's compare side BC to side EF. We have BC = 4 millimeters and EF = 12 millimeters. Therefore, we have 4/12.
Finally, let's compare side CA to side FD. We have CA = 8.5 millimeters and FD is labeled as x. Therefore, we have 8.5/x.
Now, we can set up our proportion: (10/DE) = (4/12) = (8.5/x).
To solve for x, we can cross-multiply and solve for x:
10 * x = 8.5 * 12 * 4
Now, we grab our calculators and do some math magic:
10x = 102
And finally, we divide both sides by 10 to isolate x:
x = 10.2
So, the value of x is 10.2 millimeters. Ta-da!
To find the value of x, we can use the property of similarity between two triangles. When two triangles are similar, their corresponding sides are in proportion.
We can set up a proportion using the corresponding sides of triangle ABC and triangle DEF:
AB/DE = BC/EF = CA/FD
Plug in the given values:
10/DE = 4/12 = 8.5/x
To solve for x, we'll solve the proportion for DE/EF and then multiply by x:
10/DE = 4/12
Cross-multiply:
10 * 12 = 4 * DE
120 = 4DE
Divide both sides by 4:
120/4 = DE
30 = DE
Now, we have the value of DE. To find the value of x, we can multiply DE by the ratio of FD/EF:
30 * (8.5/12) = x
30 * 0.7083 = x
x ≈ 21.25 millimeters
EF / BC = FD / CA
1.5
FD/CA = EF/BC
x/8.5 = 12/4.
X = (8.5*12)/4 =