Answered

Find expert answers and community-driven knowledge on IDNLearn.com. Get accurate and detailed answers to your questions from our knowledgeable and dedicated community members.

The distance between city A and city B is 22 miles. The distance between city B and city [tex]\(C\)[/tex] is 54 miles. The distance between city [tex]\(A\)[/tex] and city [tex]\(C\)[/tex] is 51 miles.

What type of triangle is created by the three cities?

A. An acute triangle, because [tex]\(22^2 + 54^2 \ \textgreater \ 51^2\)[/tex]

B. An acute triangle, because [tex]\(22^2 + 51^2 \ \textgreater \ 54^2\)[/tex]

C. An obtuse triangle, because [tex]\(22^2 + 54^2 \ \textgreater \ 51^2\)[/tex]

D. An obtuse triangle, because [tex]\(22^2 + 51^2 \ \textgreater \ 54^2\)[/tex]

Sagot :

GinnyAnsweridn
Let's analyze the given distances to classify the triangle formed by the cities A, B, and C.

We are given:
- The distance between city A and city B (AB) = 22 miles.
- The distance between city B and city C (BC) = 54 miles.
- The distance between city A and city C (AC) = 51 miles.

To classify the triangle based on its angles, we'll use the properties of the squares of the sides and the Pythagorean theorem:

1. If [tex]\( a^2 + b^2 > c^2 \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the sides of the triangle, the triangle is acute.
2. If [tex]\( a^2 + b^2 < c^2 \)[/tex], the triangle is obtuse.
3. If [tex]\( a^2 + b^2 = c^2 \)[/tex], the triangle is right.

Calculating the squares of each side:
- [tex]\( AB^2 = 22^2 = 484 \)[/tex]
- [tex]\( BC^2 = 54^2 = 2916 \)[/tex]
- [tex]\( AC^2 = 51^2 = 2601 \)[/tex]

Now, let's compare the sums of the squares of two sides to the square of the third side to determine the type of triangle:

Step 1: Check [tex]\( AB^2 + BC^2 \)[/tex] and compare it to [tex]\( AC^2 \)[/tex]:
- [tex]\( AB^2 + BC^2 = 484 + 2916 = 3400 \)[/tex]
- [tex]\( AC^2 = 2601 \)[/tex]

Since [tex]\( 484 + 2916 > 2601 \)[/tex]:
- [tex]\( 22^2 + 54^2 > 51^2 \)[/tex]
This comparison indicates that the triangle is obtuse.

Therefore, the triangle formed by the cities A, B, and C is an obtuse triangle. The correct statement is:

"An obtuse triangle, because [tex]\(22^2+54^2 > 51^2\)[/tex]"
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.