In triangle SCU shown below, points T and O are on SU and CU respectively. Segment OT is drawn so that angle C is congruent to angle OTU. If TU=4, OU=5, and OC=7,what is the length of ST
The Correct Answer and Explanation is :
To solve for the length of segment ( ST ) in triangle ( SCU ) with points ( T ) and ( O ) on segments ( SU ) and ( CU ), respectively, we can utilize properties of similar triangles and the concept of angle congruence.
Given:
- ( TU = 4 )
- ( OU = 5 )
- ( OC = 7 )
Since ( \angle C \cong \angle OTU ), triangles ( OCU ) and ( OTU ) are similar by the AA (Angle-Angle) similarity criterion. This means that the corresponding sides of these triangles are proportional.
Let’s denote:
- ( ST = x )
- ( SU = ST + TU = x + 4 )
- ( CU = OC + OU = 7 + 5 = 12 )
Using the property of similar triangles, we can set up the following proportion based on the corresponding sides:
[
\frac{ST}{TU} = \frac{OC}{OU}
]
Substituting the known values into the proportion:
[
\frac{x}{4} = \frac{7}{5}
]
Now, we can cross-multiply to solve for ( x ):
[
5x = 4 \cdot 7
]
[
5x = 28
]
[
x = \frac{28}{5}
]
[
x = 5.6
]
Thus, the length of segment ( ST ) is ( 5.6 ).
Explanation:
- Triangle Similarity: The key to solving this problem is recognizing the similarity between triangles ( OCU ) and ( OTU ). Since ( \angle C ) is congruent to ( \angle OTU ), and both triangles share angle ( U ), we can confidently apply the properties of similar triangles.
- Setting Up Proportions: By defining the lengths of segments in terms of ( ST ) and using known values for the other segments, we can express the relationship between the sides of the triangles through a proportion.
- Solving the Equation: After establishing the proportion, it is a matter of algebraic manipulation to find ( x ), which represents the length of ( ST ).
In conclusion, the calculated length of ( ST ) is ( 5.6 ), determined through the application of triangle similarity and proportional reasoning.