Answer:
[tex]x=14[/tex]
Skills needed: Triangle Similarity
Step-by-step explanation:
1) We are given two triangles. Within each of these triangles, there are angles with arcs.
IMPORTANT: Both the angles marked with two arcs are congruent ([tex]\cong[/tex]) to each other.
- That means also both the angles marked with one arc are [tex]\cong[/tex] to each other.
------------------------------------------------------------------------------------------------------------- The bigger triangle, [tex]\triangle{ABC}[/tex]2 angles congruent to the smaller triangle, [tex]\triangle{DE}F[/tex], which means that it is [tex]AA[/tex] similarity.
---> When two triangles are similar, that means that one triangle is just a scaled-version of the other triangle.
[tex]AA[/tex] similarity means that two angles from each triangle are congruent. This is one similarity method.
2) Since we know these triangles are similar with the angle arcs, we can use a formula to find the side length [tex]x[/tex], which is [tex]\overline{DF}[/tex]
---> We have the angle, [tex]\angle BAC[/tex], which is congruent to [tex]\angle EDF[/tex]
---> We have angle, [tex]\angle BCA[/tex], which is congruent to [tex]\angle EFD[/tex]
[tex]\frac{\text{side opposite of } \angle BAC}{\text{side opposite of }\angle EDF} = \frac{\text{side opposite of } \angle ABC}{\text{side opposite of }\angle DE\text{F}}[/tex]
(We know that the other pair of angles in the triangle are congruent to each other when given AA similarity)
The sides opposite of congruent angles are proportional to each other in a similar triangle:
[tex]\frac{24}{16}=\frac{21}{x}[/tex]
3) In order to solve this, we use cross products:
[tex]\frac{a}{b} = \frac{c}{d}, a*d=b*c[/tex]
[tex]24x=21*16 \\ 24x=336 \\ x=14[/tex]---> Here, we use cross products and isolate the variable, and get a valid solution for the variable. 14 is the answer.
