Question

Given:


. Find the length of

.

The diagram shows triangle H G I with point P on side H G and point Q on side G I. Segment P G is labeled 10, segment G Q is labeled 15, and segment Q I is labeled 24.

Responses

23
Pick the answer below.

26
Pick the answer below.

28
Pick the answer below.

30

Answer 1

To find the length of segment HI in triangle HGI, we can apply the triangle inequality theorem, but first, we'll clarify the relationships of the segments given:

1. PG = 10
2. GQ = 15
3. QI = 24

We want to find the length of HI.

Since the points P, G, Q, and I are on the triangle, we can use the information given to find HI. We know that:

HG = PG + GQ + QI (the total distance along the side of the triangle).

So, to find HG, we can add the segments:

HG = PG + GQ + QI = 10 + 15 + 24 = 49.

Now, applying triangle inequality, we also need to ensure that:

1. HI + IG > HG
2. HI + HG > IG
3. IG + HG > HI

However, since HI is directly unknown and IG must be derived, the most logical choice for HI consistent with the segments provided involves comparing distances.

Since we know that the maximum obtainable length of HI has to be equal to or less than the full segment of the triangle, the reasonable option appears to be **30**.

Therefore, the length of HI is likely **30**.