Triangle HJK is cut by line segment AB. Line segment AB goes from side HJ to side HK. The length of HA is 5.25 inches, the length of HB is 3 inches, and the length of BK is 5 inches. What is the length of line segment AJ if line segment AB is parallel to line segment JK?
Options:
A) 8 in.
B) 8.75 in.
C) 10.25 in.
D) 14 in.
Answer (2)
To find the length of line segment A J , we use the concept of similar triangles. Given that line segment A B is parallel to J K , triangles H A B and H J K are similar by the Basic Proportionality Theorem (also known as Thales' theorem).
With the triangles being similar, the corresponding sides are proportional. This means:
H J H A = HK H B
We are given:
H A = 5.25 inches
H B = 3 inches
B K = 5 inches
We need to find A J , and we know that:
H J = H A + A J
From the proportion:
5.25 + A J 5.25 = 3 + 5 3
This simplifies to: 5.25 + A J 5.25 = 8 3
Cross-multiply to solve for A J :
5.25 × 8 = 3 × ( 5.25 + A J )
42 = 15.75 + 3 A J
Subtract 15.75 from 42 :
42 − 15.75 = 3 A J
26.25 = 3 A J
Divide both sides by 3 :
A J = 3 26.25
A J = 8.75 inches
Therefore, the length of line segment A J is 8.75 inches.
The correct option is B) 8.75 in.
The length of line segment AJ is calculated to be 8.75 inches using the properties of similar triangles. Because line segment AB is parallel to JK, we can set up a proportion based on corresponding sides of the similar triangles. The correct answer is option B) 8.75 in.
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